General Topology 9630509709, 9789630509701

Table of contents :
Preface to the English Edition
Contents
1. Introduction
1.1. Sets
1.1.a. Membership
1.1.b. Subsets
1.1.c. Operations
1.1.d. Equivalence relations
1.1.e. Sets of real numbers
1.1.f. Numerical sequences
1.1.g. Countable sets
1.1.h. Well-ordering
1.1.i. Kuratowski-Zorn lemma
1.1.j. Exercises
1.2. Euclidean spaces
1.2.a. Distance
1.2.b. Convergence of a sequence of points
1.2.c. Open and closed sets
1.2.d. Dense sets
1.2.e. Theorems of Cantor, Lindelöf and Borel
1.2.f. Exercises
1.3. Metric spaces
1.3.a. Metric
1.3.b. Convergence. Open and closed sets
1.3.c. Complete metric spaces
1.3.d. Separable metric spaces
1.3.e. Distance between sets
1.3.f. Pseudo-metrics
1.3.g. Pseudo-metric spaces
1.3.h. Exercises
2. Topological spaces
2.1. The notion of topological space
2.1.a. Convergence
2.1.b. Centred systems, filter bases, filters
2.1.c. Neighbourhood structures
2.1.d. Convergence, open and closed sets in a neighbourhood space
2.1.e. Topological spaces
2.1.f. Exercises
2.2. Determination of topologies
2.2.a. Neighbourhood bases
2.2.b. Bases
2.2.c. Prescription of open or closed sets
2.2.d. Interior and closure of a set
2.2.e. The boundary of a set
2.2.f. Axiomatic remarks
2.2.g. Exercises
2.3. Comparison and restriction of topologies
2.3.a. Comparison of topologies
2.3.b. Restriction of topologies. Subspaces
2.3.c. Exercises
2.4. Convergence of filter bases
2.4.a. Insufficiency of the convergence of sequences
2.4.b. Convergence of filter bases
2.4.c. Axioms of countability
2.4.d. Examples. Metrizable spaces
2.4.e. Exercises
2.5. Separation axioms
2.5.a. Basic notions
2.5.b. T_0-spaces
2.5.c. T_1-spaces
2.5.d. T_2-spaces
2.5.e. Regular spaces
2.5.f. Normal spaces
2.5.g. Completely normal spaces
2.5.h. Exercises
2.6. Continuous mappings
2.6.a. Mappings
2.6.b. Image and inverse image of a system of sets
2.6.c. Continuous mappings
2.6.d. Homeomorphy
2.6.e. Continuous functions
2.6.f. Inverse image of topologies
2.6.g. Exercises
3. Proximity and uniform spaces
3.1. Proximity spaces
3.1.a. Proximity of a pseudo-metric space
3.1.b. Proximity spaces
3.1.c. Topology of a proximity space
3.1.d. Comparison of proximity relations
3.1.e. Restriction of proximities
3.1.f. Inverse image of proximities
3.1.g. Proximally continuous maps
3.1.h. Exercises
3.2. Uniform spaces
3.2.a. ε-Surroundings in a pseudo-metric space
3.2.b. Cartesian product of two sets
3.2.c. Uniform spaces
3.2.d. Uniformity induced by a family of pseudo-metrics
3.2.e. Proximity and topology of a uniform space
3.2.f. Comparison of uniformities
3.2.g. Restriction of uniformities
3.2.h. Inverse image of uniformities
3.2.i. Uniformly continuous maps
3.2.j. Totally bounded uniform spaces
3.2.k. Exercises
4. Completely regular spaces
4.1. Urysohn's lemma
4.1.a. Ordering in proximity and uniform spaces
4.1.b. Urysohn's lemma
4.1.c. Exercises
4.2. Completely regular spaces
4.2.a. The notion of a completely regular space
4.2.b. Families of functions
4.2.c. Inducing of topologies and proximities by function families
4.2.d. Inducing of uniformities by families of pseudo-metrics
4.2.e. Characterization by means of subbases
4.2.f. Exercises
5. Complete and compact spaces
5.1. Complete uniform spaces
5.1.a. Cauchy filter bases
5.1.b. Complete uniform spaces
5.1.c. Exercises
5.2. Compact proximity spaces
5.2.a. Compressed filter bases
5.2.b. Ultrafilters
5.2.c. Compact proximity spaces
5.2.d. Cluster points of filter bases
5.2.e. Exercises
5.3. Compact topological spaces
5.3.a. Various characterizations
5.3.b. Properties of compact spaces and sets
5.3.c. Countably compact spaces
5.3.d. Sequentially compact spaces
5.3.e. Locally compact spaces
5.3.f. Rim-compact spaces
5.3.g. Exercises
6. Extensions of spaces
6.1. Extensions of topological spaces
6.1.a. The notion of extension
6.1.b. Strict extensions
6.1.c. The Alexandroff compactification
6.1.d. \mathfrak{H}-filters
6.1.e. Wallman-type compactifications
6.1.f. Wallman compactification
6.1.g. Freudenthal compactification
6.1.h. H-closed extensions
6.1.i. Exercises
6.2. Extension of mappings
6.2.a. Extension of continuous mappings
6.2.b. Extension of uniformly continuous mappings
6.2.c. Extension of proximally continuous mappings
6.2.d. Exercises
6.3. Extensions of uniform spaces
6.3.a. Round filters
6.3.b. Extensions of a uniform space
6.3.c. Completion of a uniform space
6.3.d. Exercises
6.4. Extensions of proximity spaces
6.4.a. Extensions of a proximity space
6.4.b. Compactification of a proximity space
6.4.c. Compactifications of a completely regular space
6.4.d. Čech-Stone compactification
6.4.e. Real-compact spaces
6.4.f. Hewitt real-compactification
6.4.g. Exercises
7. Product and quotient spaces
7.1. The product of topological spaces
7.1.a. Projective generation
7.1.b. Cartesian product of sets
7.1.c. The product of topological spaces
7.1.d. The product of compact spaces
7.1.e. Embedding theorems
7.1.f. Exercises
7.2. The product of proximity spaces
7.2.a. Projective generation of proximities
7.2.b. The product of proximity spaces
7.2.c. Embedding theorems
7.2.d. Exercises
7.3. The product of uniform spaces
7.3.a. Projective generation of uniformities
7.3.b. The product of uniform spaces
7.3.c. Embedding theorems
7.3.d. Square of uniformities
7.3.e. Exercises
7.4. Quotient spaces
7.4.a. Inductive generation of topologies
7.4.b. Quotient topologies
7.4.c. Quotient spaces
7.4.d. Quotient spaces of proximity spaces
7.4.e. Quotient spaces of uniform spaces
7.4.f. Exercises
8. Paracompact spaces
8.1. Divisible spaces
8.1.a. Neighbourhoods of the diagonal
8.1.b. Multinormal spaces
8.1.c. Equicontinuous functions
8.1.d. Further characterizations
8.1.e. Exercises
8.2. Fully normal spaces
8.2.a. Refinements and star-refinements of a system of sets
8.2.b. Fully normal spaces
8.2.c. Ultracomplete spaces
8.2.d. Exercises
8.3. Paracompact spaces
8.3.a. Locally finite systems of sets
8.3.b. Paracompact spaces
8.3.c. Partitions of unity
8.3.d. Equivalent characterizations
8.3.e. Examples of paracompact spaces
8.3.f. Product theorems
8.3.g. Metacompact spaces
8.3.h. Continuous closed images of paracompact spaces
8.3.i. Countably paracompact spaces
8.3.j. Strongly paracompact spaces
8.3.k. Exercises
8.4. Metrization theorems
8.4.a. Regular and point-regular bases
8.4.b. Perfectly normal spaces
8.4.c. Metrization conditions
8.4.d. Applications
8.4.e. Metrizability of proximity spaces
8.4.f. Continuous closed images of metrizable spaces
8.4.g. Embedding into product spaces
8.4.h. Exercises
9. Baire spaces
9.1. Rare and meagre sets
9.1.a. Rare sets
9.1.b. Meagre sets
9.1.c. Exercises
9.2. Baire spaces
9.2.a. Baire spaces
9.2.b. Base-compact and hypocompact spaces
9.2.c. Product theorems
9.2.d. Applications
9.2.e. Čech-complete spaces
9.2.f. Completely metrizable spaces
9.2.g. Exercises
10. Connected spaces
10.1. Connected sets
10.1.a. Separated partitions
10.1.b. Connected sets
10.1.c. Operations
10.1.d. Components
10.1.e. Continua
10.1.f. Exercises
10.2. Locally connected spaces
10.2.a. Local connectedness
10.2.b. Operations
10.2.c. Arcs
10.2.d. Exercises
10.3. Arcwise connected spaces
10.3.a. Arcwise joined points
10.3.b. Arcwise connected sets
10.3.c. Locally arcwise connected sets
10.3.d. Chains
10.3.e. Locally connected complete metric spaces
10.3.f. Exercises
10.4. Locally connected continua
10.4.a. Covering with continua
10.4.b. Continuous images
10.4.c. Exercises
11. Topological groups
11.1. Groups
11.1.a. The notion of group
11.1.b. Examples
11.1.c. Multiplication of sets
11.1.d. Translations
11.1.e. Subgroups
11.1.f. Homomorphisms
11.1.g. Exercises
11.2. Topological groups
11.2.a. The notion of topological group
11.2.b. Neighbourhood bases of e
11.2.c. Consequences
11.2.d. Subgroups
11.2.e. Homomorphisms
11.2.f. Exercises
11.3. Complete groups
11.3.a. Admissible uniformities
11.3.b. The admissibility of left and right uniformities
11.3.c. Bilateral uniformity
11.3.d. The completion of a topological group
11.3.e. Locally compact groups
11.3.f. Exercises
Literature
Subject index
Notations

Citation preview

GENERAL TOPOLOGY AKOS CSASZAR CORRESPONDING MEMBER OF THE HUNGARIAN ACADEMY OF SCIENCES PROFESSOR OF MATHEMATICS AT THE EOTVOS LORAND UNIVERSITY

ADAM HILGER LTD BRISTOL

GENERAL TOPOLOGY

This Monograph was published in Hungary as Vol. 9 in the series of DISQUISITIONES MATHEMATICAE HUNGARICAE

Translated by

MRS A. CSASZAR

© Akademiai Kiad6, Budapest 1978

Published by Adam Hilger Ltd A company now owned by The Institute of Physics Techno House, Redcliffe Way, Bristol BSl 6NX, England

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN O 85274 275 4

Published as a co-edition of Adam Hilger Ltd, Bristol and Akademiai. Kiad6, Budapest Printed in Htm'gary

To my wife

J '·

PREFACE TO THE ENGLISH EDITION

The literature of general topology is enriched each year by several hundreds of papers containing new scientific results and besides these, by some new books or new editions of older books. Some of them deal with a specific domain of this discipline and therefore need certain preliminaries of topology, but the majority treat the subject starting from basic principles. This phenomenon is explained by the fact that one of the characteristic features of modern mathematics is the penetration of topological methods into many chapters of mathematics, first of all into analysis and geometry. Hence it is desirable that students interested in general topology can embark on this subject in. a rather early phase of their studies. From a purely logical point of view it would be possible to present general topology in a form disregarding any previous knowledge in mathematics, by beginning, after the enumeration of the background from set theory, with the basic definitions and axioms. However, I am convinced that a treatment of this kind is an abuse of the possibilities of the axiomatic method which can produce a formal understanding but cannot furnish a real insight into the importance of the concepts. Indeed, there is no doubt that the abstract theories characterizing. modern mathematics are important only because their general concepts include a series of interesting special cases; their general statements, applied to conc:r:ete examples, imply interesting conclusions, and I am firmly convinced that the presentation of such a theory is useful only if it includes the necessary preparation for the procedure of abstraction leading to the definitions, the explanation of the special cases contained in them, and the applications resulting from the theorems. However, in order to reach the concepts of general topology in a natural way, starting from concrete notions, and to illustrate the definitions by non-trivial examples, one needs some background in analysis; the larger this background, the richer is the illustrative material available in presenting the theory, but of course then the possibility of getting acquainted with topology is transferred to a later phase of studies in mathematics. After considering all this, I decided to postulate for the reader a knowledge corresponding to the material of a first year undergraduate course in analysis. Instead of presuming "nothing else than ability in abstract thinking", I assume, to the extent indicated above, acquaintance with elementary analysis. On the other hand, background from analysis, algebra, or set theory, not familiar to a student having finished the first year at university, is not used without a detailed explanation at the corresponding place in the text.

8

PREFACE

This limitation of mathematical background implies that l do not use the concept of the cardinality of a set - except the distinction of finite, countable, and uncountable sets - as well as the theory of infinite cardinal and ordinal numbers. The Kuratowski-Zorn lemma and the Zermelo theorem, stated without proof, give however the possibility of presenting every essential concept and result of general topology, except, of course, the concepts based on the notion of cardinality (e.g. the weight of a topological space) and some more intricate counter-examples. An expert reader can judge whether this avoidance of deeper methods of set theory essentially truncates the material or not. According to the introductory character of the book the purpose of the exercises at the end of each section is to give the reader the possibility of controlling the understanding of the material by enlightening the content of a definition or a theorem by means of special cases and counter-examples. I do not have the illusion (in contrast to the authors of many similar books) that the reader will be able to discover alone each further result of general topology; accordingly every exercise (except those which are straightforward) is divided into smaller units and often the · essential thoughts necessary to the solution are given in brackets. Another aid to the beginner is the clear presentation of the logical structure of longer proofs. This is done by means of the symbols and ; the latter is expressed in the text by the usual abbreviation "iff". The symbol I shows the end of a proof; ifit is put at the end of a statement, then the latter is obvious in view of the corresponding definitions or the preceding statements. According to the character of the book, references to the literature are omitted. Instead there is a list of textbooks and monographs on general topology, rather large without being complete. It does not contain works in languages other than English, French, or German; new editions are mentioned only if their language differs from that of the first one, or if they were essentially enlarged. The works contained in this list give a rich possibility of getting deeper knowledge in general topology; a part of them contains a large list of papers on the subject. The introductory character of the book was emphasized above several times. In apparent contradiction to this, I should like to point out that I tried to present to the reader the up-to-date features of general topology and to give a survey on some fairly recent theories. In particular, I present some types of topological structures other than topologies in some extent and I point out their role in the study of various questions of general topology. The first Hungarian edition of this book (Bevezetes az altalanos topologiaba) appeared in 1970. The English edition was enlarged by the treatment of some new subjects, especially in Chapters 4, 8 and 9, and many new exercises were added. It is my duty to express my sincerest gratitude to my colleagues M. Bognar, S. Gacsalyi, J. Ger lits, P. Hamburger, I. Juhasz, for their aid in preparing. the manuscript. Some proofs and a series of exercises are based on their ideas.

=

A. Csaszar

CONTENTS

1. Introduction

15

1.1. Sets I.I.a. 1.1.b. 1.1.c. 1.1.d. 1.1.e. 1.1.f. I.Lg. 1.1.h.

15 15 16 16 17 17 21 21

Membership Subsets . . Operations Equivalence relations Sets of real numbers Numerical sequences Countable sets Well-ordering 1.1.L Kuratowski-Zorn lemma 1.1.j. Exercises 1.2. Euclidean spaces . . . . . . 1.2.a. Distance . . . . . . 1.2.b. Convergence of a sequence of points 1.2.c. Open and closed sets . . . . . . 1.2.d. Dense sets , ...... . 1.2.e. Theorems of Cantor, Lindelof and Borel 1.2.f. Exercises 1.3. Metric spaces . . . . . . ,,: . . . . . . 1.3.a. Metric . . . . . . . . . . . . . 1.3.b. Convergence. Open and closed sets 1.3.c. Complete metric spaces . 1.3.d. Separable metric spaces 1.3.e. Distance between sets 1.3.f. Pseudo-metrics 1.3.g. Pseudo-metric spaces 1.3.h. Exercises . . . . . . 2. Topological spaces 2.1. The notion of topological space 2.1.a. Convergence ..... 2.1. b. Centred systems, filter bases, filters 2.1.c. Neighbourhood structures 2.1.d. Convergence, open and closed sets in a neighbourhood space 2.1.e. Topological spaces 2.1.f. Exercises . • . . . . 2.2. Determination of topologies 2.2.a. Neighbourhood bases 2.2. b. Bases . . . . . . .

24 24 25 26 26

29 33 36 37 39 40 40 43

44 45

46 49 50 50 53 53 53 55 59 61 62

64 66 66 67

10

CONTENTS

2.3.

2.4.

2.5.

2.6.

2.2.c. Prescription of open or closed sets 2.2.d. Interior and closure of a set 2.2.e. The boundary of a set 2.2.f. Axiomatic remarks 2.2.g. Exercises . . . . . . . . . Comparison and restriction of topologies 2.3.a. Comparison of topologies 2.3.b. Restriction of topologies. Subspaces 2.3.c. Exercises . . . . . • . . . . . . . Convergence of filter bases . . . . . . . 2.4.a. Insufficiency of the convergence of sequences 2.4.b. Convergence of filter bases 2.4.c. Axioms of countability 2.4.d. Examples. Metrizable spaces 2.4.e. Exercises Separation axioms 2.5.a. Basic notions 2.5. b. T0 -spaces 2.5.c. Ti-spaces 2.5.d. T2-spaces 2.5.e. Regular spaces 2.5.f. Normal spaces . 2.5.g. Completely normal spaces 2.5.h. Exercises Continuous mappings 2.6.a. Mappings 2.6.b. Image and inverse image of a system of sets 2.6.c. Continuous mappings 2.6.d. Homeomorphy • . . . . . 2.6.e: Continuous functions 2.6.f. Inverse image of topologies 2.6.g. Exercises . . . . .

3. Proximity and uniform spaces 3.1. Proximity spaces . . . • . . . 3.1.a. Proximity of a pseudo-metric space 3.1.b. Proximity spaces ...... . 3.1.c. Topology of a proximity space· 3.1.d. Comparison of proximity relations 3.1.e. Restriction of proximities 3.1.f. Inverse image of proximities 3.1.g. Proximally continuous maps 3.1.h. Exercises 3.2. Uniform spaces . . . . . . . . . 3.2,a. e-Surroundings in a pseudo-metric space 3.2.b. Cartesian product of two sets 3.2.c. Uniform spaces . . . . . . . . . 3.2.d. Uniformity induced by a family of pseudo-metrics 3.2.e. Proximity and topology of a uniform space 3.2.f. Comparison of uniformities 3.2.g. Restriction of uniformities 3.2.h. Inverse image of uniformities 3.2.i. Uniformly continuous maps

69 70 72 73 73

75 75 77

so

82 82 83

85 87

89 91 91 92

93 94

96 97

98 101 102 102 105 107 111 112 115

118 121 121 121 122 124 125 129 130

131 134 136 136 137 139

140 141 142 144

146

147

CONTENTS

3.2.j. Totally bounded uniform spaces · 3.2.k. Exercises . . . . . . . . 4. Completely regular spaces 4.1. Urysohn's lemma 4.1.a. Ordering in proximity and uniform spaces 4.1.b. Urysohn's lemma 4.l.c. Exercises . . . . . . . . . . . • . 4.2. ·Completely regular spaces . . . . . . . 4.2.a. The notion of a completely regular space 4.2.b. Families of functions ...•..... 4.2.c. Inducing of topologies and proximities by function families 4.2.d. Inducing of uniformities by families of pseudo-metrics 4.2.e. Characterization by means of subbases 4.2.f. Exercises . . . • . . . . . . . . . . . . 5. Complete and compact spaces

5.1. Complete uniform spaces, 5.1.a. Cauchy filter bases 5.Lb. Complete uniform spaces 5.1.c. Exercises . . . . . . 5.2. Compact proximity spaces 5.2.a. Compressed filter bases 5.2.b. Ultrafilters . . . . . 5.2.c.. Compact proximity spaces 5.2.d. Cluster points of filter bases 5.2.e. Exercises . . . . . . . 5.3. Compact topological spaces 5.3.a. Various characterizations . . 5.3.b. Properties of compact spaces and sets 5.3.c. Countably compact spaces 5.3.d. Sequentially compact spaces 5.3:e. Locally compact spaces 5.3.f. Rim-compact spaces 5.3.g. Exercises . . . . . . ·6. Extensions of spaces 6.1. Extensions of topological spaces · 6.1.a. The notion of extension 6.1.b. Strict extensions . . . . 6.1.c. The Alexandroff compactification '6.1.d. ~-filters . . . . . . . . . . 6.1.e. Wallman~type compactifications 6.1.f. Wallman compactification 6.1.g. Freudenthal compactification 6.1.h. H-closed extensions 6.1.i. Exercises . . . . . . . . . 6.2. Extension of mappings . . . . . . 6.2.a. Extension of continuous mappings 6.2.b. Extension of uniformly continuous mappings 6.2.c. Extension of proximally continuous mappings

11

152 157 160 160 160 161 165 166 166 168 170 171 174 177 182 182 182 184 185 186 186 187

189 190

192 193 193 195

199 202 203 204 208

213 213 213

215 219 221 225 235 237 238

241 245 245 246 248

12

CONTENTS

6.2.d. Exercises . . . . . . 6.3. Extensions of uniform spaces 6.3.a. Round filters 6.3.b. Extensions of a uniform space 6.3.c. Completion of a uniform space 6.3.d. Exercises . . . . . . . . . . 6.4. Extensions of proximity spaces 6.4.a. Extensions of a proximity space 6.4.b. Compactification of a proximity space 6.4.c. Compactifications of a completely regular space 6.4.d. Cech-Stone compactification 6.4.e. Real-compact spaces . . . 6.4.f. Hewitt real-compactification 6.4.g. Exercises 7. Product and quotient spaces 7.1. The product of topological spaces 7.1.a. Projective generation 7.1.b. Cartesian product of sets 7.1.c. The product of topological spaces 7.1.d. The product of compact spaces 7.1.e. Embedding theorems 7. l.f. Exercises . . . . . . . . . . 7.2. The product of proximity spaces 7.2.a. Projective generation of proximities 7.2.b. The product of proximity spaces 7.2.c. Embedding theorems 7.2.d. Exercises . . . . . . . . . . . 7.3. The product of uniform spaces 7.3.a. Projective generation of uniformities 7.3.b. The product of uniform spaces 7.3.c. Embedding theorems 7.3.d. Square of uniformities 7.3.e. Exercises . . . . . 7.4. Quotient spaces 7.4.a. Inductive generation of topologies 7.4.b. Quotient topologies . . . . . . . 7.4.c. Quotient spaces . . . . . . . 7.4.d. Quotient spaces of proximity spaces 7.4.e. Quotient spaces of uniform spaces 7.4.f. Exercises . . . . . . . 8. Paracompact spaces 8.1. Divisible spaces 8.1.a. Neighbourhoods of the diagonal 8.1.b. Multinormal spaces 8.1.c. Equicontinuous functions 8.1.d. Further characterizations 8.1.e. Exercises . . . . . . . 8.2. Fully normal spaces 8.2.a. Refinements and star-refinements of a system of sets . . . . . . . . . . . . . 8.2.b. Fully normal spaces

248 249 249 252 255 257 258 258 259

261 264 266 273 275 280 280 280 282 283 287 290 297 301 301 302 304 304 306 306 308 311

312 315 317

317 318 320 322 325 327 334 334 334 335 336 337

338 339 339 340

CONTENTS

8.2.c. Ultracomplete spaces 8.2.d. Exercises . . . . . 8.3. Paracompact spaces 8.3.a. Locally finite systems. of sets 8.3.b. Paracompact spaces 8.3.c. Partitions of unity 8.3.d. Equivalent characterizations 8.3.e. Examples of paracompact spaces ·8.3.f. Product theorems . . . . . . . 8.3.g. Metacompact spaces . . . . . • 8.3.h. Continuous closed images ofparacompact spaces 8.3.i. Countably paracompact spaces 8.3.j. Strongly paracompact spaces 8.3.k. Exercises . . . . . . . . . . 8.4. Metrization theorems . . . . . . . 8.4. a. Regular and point-regular bases 8.4. b. Perfectly normal spaces 8Ac. Metrization conditions . . . . . 8.4.d. Applications . . . . . . . . . 8.4.e. Metrizability of proximity spaces 8.4.f. Continuous closed images of metrizable spaces 8.4.g. Embedding into product spaces 8.4.h. Exercises . . . . . . . . . . . . . . • . . 9. Baire spaces 9.1. Rare and meagre sets 9.1.a. Rare sets 9.1.b. Meagre sets 9.1.c. Exercises 9.2. Baire spaces 9.2.a. Baire spaces 9.2.b. Base-compact and hypocompact spaces 9.2.c. Product theorems 9.2.d. Applications . . . . . . . 9.2.e. Cech-complete spaces 9.2.f. Completely metrizable spaces 9.2.g. Exercises . . . . . . . . . 10. Connected spaces 10.1. Connected sets 10.1.a. Separated partitions 10.1.b. Connected sets 10.1.c. Operations 10.1.d. Components 10.1.e. Continua 10.1.f. Exercises. . 10.2. Locally connected spaces 10.2.a. Local connectedness 10.2.b. -Operations 10.2.c. Arcs 10.2.d. Exercises

13 341 343 344 344 346 347 348 349 351 354 357 .360 365 367 371 371 372 372

376 377

378 380 381 384 384 384 385 386 387 387 388 392 393 397 ,399 402 407 407 407 408 409 411 411

412 414 414 415 416 420

14

CONTENTS

10.3. Arcwise connected spaces 10.3.a. Arcwise joined points 10.3.b. Arcwise connected sets 10.3.c. Locally arcwise connected sets 10.3.d. Chains . . . . . . . . . . . 10.3.e. Locally connected complete metric spaces 10.3.f. Exercises . . . . . . 10.4. Locally connected continua 10.4.a. Covering with continua 10.4.b. Continuous images 10.4.c. Exercises . . . . . . 11. Topological groups 11.1. Groups ..... . 11.1.a. The notion of group 11.1.b. Examples . . . . . 11.1.c. Multiplication of sets 11.1.d. Translations 11.1.e. Subgroups 11.1.f. Homomorphisms 11.1.g. Exercises 11.2. Topological groups . . 11.2.a. The notion of topological group 11.2.b. Neighbourhood bases of e 11.2.c. Consequences 11.2.d. Subgroups 11.2.e. Homomorphisms 11.2.f. Exercises 11.3. Complete groups . . . . 11.3.a. Admissible uniformities 11.3.b, The admissibility ofJeft and right uniformities 11.3.c. Bilateral uniformity . . . . . . . . 11.3.d. The completion of a topological group 11.3.e. Locally compact groups 1l. 3.f. Exercises

423 423 424 424 425 426 428 430 430 431 435 437 437 437 438 439 440 440 441 443 445 445 446 449 451 453 453 456 456 457 461 463 466 467

Literature

471

Subject index

475

Notations

482

1. INTRODUCTION

1.1. SETS 1.1.a. Membership. In every-day life and in mathematics too we frequently have to deal with notions representing the "collection" of certain things which "belong" to the collection in question, while other things do not belong to it; e.g. a class in a school is the collection of the pupils enumerated in the register; these pupils belong to the class, other things (other persons, animals, objects, plane figures, etc.) do not belong to the class. To the circumference drawn in the plane S around the pointp of S with radius r = 2 belong thos'e points of S whose distance from p is equal to 2; other things (other points of the plane S, points not belonging to the plane S, pupils, missiles, etc.) do not belong to the circumference. The real numbers greater than Oand smaller than 1 belong to the interval (0, 1); other things (further real numbers, books, plants, etc.) do not belong to it. We understand by a set a notion of the type outlined above, i.e. for complete determination of the set we must be able to decide for each object whether it belongs to the set or not. Those objects that belong to it are called elements of the set. E.g. the elements of the interval (0, 1) are those real numbers which are greater than O and smaller than 1. We indicate by x E A that xis an element of the set A, and by x E£ A that x is not an element of A. Consequently, the sets A and B are equal (we denote this by A = B) if both x EA implies x EB and x EB implies x EA. Thus, ½E (0, 1), but -2 Et (0, 1). The f~ct that the sets A and Bare not equal is denoted by A =/: B. It may occur that a set has no elements at all. There is a unique set with this property; it is called the empty set and is denoted by 0. Accordingly, x Et 0 for any x. The set whose only elements are a, b, c, dis denoted by {a, b, c, d}. We denote in the same way a set whose elements cannot be finitely enumerated, but the notation refers unambiguously to them; e.g. the set of even natural numbers can be denoted by the symbol {2, 4, 6, 8, ... }. Let P(x) be a statement which can be true or false, according to the choice of x; {x: P(x)} then denotes the set of those x for which P(x) is true. E.g. {x: 0 < < x < 1} denotes the interval (0, 1), and if p(p, x). denotes the distance between the points p and x of the plane S, {x: x ES, p(p, x) = 2} denotes the circumference drawn in the plane S around the point p with the radius 2. This notation will be frequently used in a simplified form; e.g. {x2 + 1: 0 ~ x ~ l} will be written in~tead of {y: y = x 2 + 1, 0 ~ x ~ 1}, and {x1: i = 1, 2, ... , 10} instead of {x: x = x 1, i = 1, 2, ... , 10}.

16

1.1.

(1.1.1)

SETS

To make our style more vivid we shall use other synonymous terms instead of the word "set": collection, system, class, family, etc. 1.1.b. Subsets. The set A is said to be a subset of the set B if each element of A belongs to B - in symbols: A c B or B :::, A. Consequently, the empty set is .a subset of every set. Any set is its own subset; the subsets different from it are called proper subsets. Thus, the set of the integers divisible by 4 is a subset of the set of the even integers; the interval (0, 1) is a subset of the interval [O, 2]. * (I.I.I) ForanysetsA, B, C, (a) Ac B, Be C implies Ac C; (b) A c B and BC A iff A = B. I 1.1.c. Operations. We understand by the union of the sets A and B, and denote · byA u B, the set of which x is an element iff x EA or x EB (including the case when x is an element of both A and B). E.g. (0, 3) u (2, 4) = {O, 4). More generally, if a set A 1 is"'assigned to each element iof an index set I, then we understand by the union of the sets Ai, denoted by the symbol

the set to which x belongs iffthere is an i EI such that x EA 1• n

If I denotes the set of integers from m to n, the notation

n

UA 1 or UA 1 will i=m

m Cl()

be used; if I denotes the set of integers not less than m, the notation

Cl()

UA 1 or UA 1

i=m

m

will be used. The intersection of the sets A and B is the set of which x is an element iff x EA and x EB simultaneously; we denote it by An B. E.g. (0, 3) n (2, 4) = (2, 3). More generally,

denotes the set of aH x with .th.e property x EA 1 for every i E I. In the case of .

n

special index sets we use here too the notation ·

oo

nA or nA 1

m

1•

m

If A n B = 0, we say that the sets A and B are disjoint. More generally, the family .of sets ·{A1 : i EJ} is. said to be disjoint if i ,fa j implies Ai n Ai = 0. The associative law is valid as follows: ,

(L 1.2). .if I = Ulj,

Bi =

(Ll.3) If I

7

Ulj, cj

UA1, then UA, = U~- I iEit

iEJ .

.iEI

= nA,, t.hen nA1 =

iEJ

· * For definition, see 1:1;e.

iEIJ

iEI

jEJ

n ½· I

}EJ

(l.1.7)

1.1.b.

17

SUBSETS

Furthermore the distributive law is valid too: (1.1.4)

(U A;) n (U Bj) = U {A; n B/ i EI, j E J}. I iEI

jEJ

The difference of the sets A and B, denoted by A - B, is the set of which x is an element iff x EA but x ~ B. E.g. (0, 3) - (2, 4) = (0, 2]. The following de Morgan identities are valid:

(1.l.5)(a) A -

n(A nB; = uCA iEI U B; =

iU

Cb) A -

B;);

iEI

iEI

BJ

1

The following distributive law is also true: (1.1.6) (A - B) n C = (A n C) - (B n C).

I

If B c A, then A - B is called the complement or complementary set of B with respect to A.

1.1.d. Equivalence relations. Suppose that the set A is the union of disjoint subsets: A =

U A;, A; n A1 = 0 if i i=

j,

iEl

and, for x, y E A, x y means that there exists an index i E I for which x E A;, y E A;. In this case, the symbol ,---..; denotes an equivalence relation on the set A, because it is obvious that the relation ,.__,. is reflexive (x ,.__,. y for x = y), symmetric (x ,.__,. y implies y ,-__,, x) and transitive (x ,-__,, y, y rov z implies x ,.__,. z). Generally we speak of an equivalence relation on a set A if there is given a relation ,-__,, between the elements of A which is reflexive, symmetric, and transitive in the above sense. E.g. if A denotes the set of all triangles of the plane, and x ,-__,, y means that the triangle xis similar to the triangle y, we get an equivalence relation. The above connection between partitions of A into disjoint subsets and equivalence relations on A is reciprocal: r-.J

(1.1. 7) Let ,.__,. be an equivalence relation on A. If A(x) denotes, for x EA, the set of those elements y EA for which x ,.__,. y, we call A(x) the equivalence class associated with x. The distinct equivalence classes give a partition of A into disjoint subsets. Proof. By the reflexivity, x E A(x); therefore the union of the equivalence classes is A. Two distinct equivalence classes are disjoint, since z E A(x) n A(y) implies x z and y z, hence z ,.__,. y, and so x y holds by transitivity. Now u E A(x) implies x u, and this, together with y x, implies y ,.__,. u, hence u E A(y). Thus A(x) c A(y) and similarly A(y) c A(x); consequently A(x) = = A(y). I r-.J

r-.J

r-.J

r-.J

r-.J

1.1.e. Sets of real numbers. We now sam;narize the well-known properties of real numbers to be used later, as well as the concepts and notations related to them (some of them have occurred already in our examples). 2 Akos Csaszar

1.1.

18

(l.1.8)

SETS

We denote by R the set of all real numbers. There is defined -among them the operation of addition which assigns the sum x + y to the numbers x ER, y ER and fulfils the commutative and associative laws: x

+y

= y

+

x, (x

+ y) + z. =

x

+ (y + z) .

There exists in R an element, denoted by 0, such that x

+0 = x

for all x E R,

and; for each x ER, there exists a y ER such that X

+ y = O;

in this case we write y = -x. We denote the sum x

+ (-y)

by the symbol

x-y.

The operation of multiplication is also defined in R, assigni~g to x ER and y

·

ER the product xy. It is commutative: xy

= yx

associative: (xy)z

= x(yz)

and distributive: x(y

+ z) =

xy

+ xz.

There exists an element of R, distinct from 0, denoted by 1, for which

x · 1 = x for x E R, and for every x E R, x =I: 0, there exists a y E R such that xy

= 1;

. t h"1s case we wnte . y = -'1 . m X

Instead of the product x _!__ , we use the notation~ (y =I: 0). y y The numbers resulting from the number 1 by iterated addition: 1

+1=

2, 2

+1=

3, 3

+1=

4, ...

are the natural numbers, whose set will be denoted by N. The natural numbers, 0, and the numbers of the form -n (n EN) together constitute the set of integers, denoted by Z. The numbers which can be written in. the form ~ (n and m are ·-

m

(1.1.8)

1.1.e,

SETS. OF REAL NUMBERS

19

integers, m ,f= 0) are called rational numbers; their set is denoted by Q. The elements of the set R - Q are the irrational numbers. The set R is an ordered set. That is to say, a relation, denoted by y means the same as y < x; x ~ y or y ~ x means that either x = y or x < y is true. The number x E R is said to be positive if x > 0, and negative if x < 0. The ordering of real numbers is connected with the operations of addition and multiplication, for the sum of two positive numbers is positive, the product of two positive or two negative numbers is positive, and the product of a positive and a negative number is negative. If a, b E R, a < l;J, then the sets (a, b) = {x: a [a, b] = {x: a [a, b) = {x: a (a, b] = {x: a

< x < b}, b}, b},

~ x ~ ~ x < < x ~

b}

are called open interval, closed interval, interval closed on the left, interval closed on the right, respectively. We use the above symbols too when a = b; then [a, a] = {a}, the other three intervals being empty. Furthermore, we use the notations

(a, +cxi) = {x: x> a}, [a, + cxi) = {x: x ~ a}, (-cxi, a)= {x: x < a}, (-cxi, a]= {x: x ~ a}, (-cxi, +cxi) = R for a ER. These' sets are called infinite intervals. The interval [O, 1] will be denoted by I. We understand by an upper bound of the set 0 ¢.Ac R a number k ER such that x EA implies x ~ k; a lower bound of A is a number k ER for which x E A implies x ~ k. The set A is bounded above or below if it has an upper or a lower bound; it is bounded whenever it is bounded both above and below. An essential property of the set R is that, among the upper bounds of any bounded subset A of R (A ,f= 0), there is a smallest; this is called the least upper bound of A and denoted by sup A.* In the same way, among the lower bounds of

* 'sup' is 2*

an abbreviation for 'supremum' and 'inf' for infimum.

20

1.1.

(l.1.9)

SETS

a bounded set A =I= 0, there exists a greatest; this is the greatest lower bound of A, denoted by inf A. ffthere is a greatest (smallest) among the elements of A, sup A (inf A) is equal to this; what is not obvious is the existence of sup A (inf A) in the case when A has no greatest (smallest) element. It can be seen easily that this property of R is not a consequence of its other properties enumerated above, for the latter are all fulfilled by the set Q, but a subset of Q, bounded above, does not always possess a least upper. bound within Q. · The usual and well-known laws of manipulation with equalities and inequalities can be deduced from the properties of real numbers enumerated above; we do not detail them now. As an application, we shall present the proofs of some frequently used statements. (1.1.9) The set N is not bounded above. Proof. If h = sup N existed, then h - 1 woµld not be an upper bound of N; hence there would be an n E N such that h - 1 < n. Hence h < n + 1 E N, in contradiction to the definition of h. I (1.1.10) inf { ~: nEN}=o. Proof. In any case, 0 is a lower bound of the set A = { ~: n E NJ. If h > 0

1 were another lower bound of A, then -;; hold, in contradiction to (1.1.9).

~

· . .1 h for all n E N, hence n ~ h would

I

(Ll.11) 1f a, b ER, a < b, then there is a q E Q such that a < q < b. 1 there Proof. By (1.1.10), there is n EN such that -n < b - a, and then by (1:1.9) . are p, q E N satisfying -nb < p, nb < q. Consider the largest element m of the finite set M = { ~p, -p + 1, ... , -1, 0, 1, ... , q - 1, q} n {x: x < nb}

· which is non-empty since - p E M. Then clearly m < q and m < b. Now !!!:.... ~ a n n m + 1 would imply - - - 0, there is an n0 E N such that [ an - am I < e for n, m

~

n0 •

I

1.1.g. Countable sets. If A is an arbitrary non-empty set, then we get a sequence in A (as a generalization of numerical sequences) by assigning an element anE A to each number n EN. We can extend to this case the notation (an) and the expressions "member of a sequence" and "subsequence". Let

t

Nm={n:nEN,n~m}

form EN; we obtain a sequence of m members or briefly an m-sequence in A by assigning elements a11 EA to the numbers n E Nm; this sequence is denoted by (a1 , . . . , am). A finite sequence in A is a sequence of m members in A for some

mEN

,

The set A is said to be finite if either A = 0 or there is a finite sequence such that its members are precisely the elements of A, i.e. for which (if the sequence is of

1.1.

(1 1.16)

SETS

m members)

The non-finite sets are saicl to be infinite. A is countable if either A = 0 or there is a sequence with

(1.1.16) Each finite set is countable. Proof. If A = {an: n E Nm}, let bn = an whenever n ~ m and bn = a1 if

n > m. ThenA = {bn: n EN},

I

The countable but not finite sets are called countably infinite. An example is N itself: (1.1.17) The set N is countably infinite. Proof. N is countable since N = {an: n E N} for an = n. If N = {bn: n E Nm} were true for some m 'E N, let c1 = bi, and let c2 be the greater of the numbers c1

and b2, c3 the greater of the numbers c2 and b3 and so on. Obviously bn ~ cm for n E Nm• so that Cm + 1 E N cannot occur among the members of the sequence (b1, .. , , bmr I (1.1.18) Each subset of a finite set is finite. Proof. If A = {an: n E Nm}, B c A and B =fa

0, let no E Nm be such that an.EB

and define bn by

We can prove in the same way: (1.1.19) Every subset of a countable set is countable,

I

m

(1.1.20) If An is a.finite set for n E Nm, then

UAn is.finite as well. 1

Proof. We can suppose that An =fa 0 for every n E Nm since, if at \east one

0, then it can be put in place of the empty An without changing the m union U An; on the other hand, if every An = 0, then the statement is obvious. 1 m Let An = {an1: i E NPn}, q == L Pn; If the elements b1, •• :, bP, are identical An.

=fa

n=l

with the elements a 11, ••• , a1P,' and similarly bp,+l• ... , bP,+P, are identical with a 21, ••• , a2P,' and so on until finally the elements bp,+ ... +Pm-t+1' ... , bP,+P,+, .. +Pm-t+Pm are identical with the elements am 1, •• • , ampm• then obviously

(1.1.25)

1.1.g.

23

COUNTABLE SETS

In the same way it can be proved that (1.1.21)

If An

00

is a finite set for each n E N, then

LJ An

is countable.

I

1

(1.1.22)

If An is countable for

00

each n EN, then

U An is countable. 1

Proof. We may again suppose that An =I= 0 for each n E N. Let An = {ani: i EN}, and furthermore

BP = {an;: n EN, i EN, n

+i=p

(p = 2, 3, ... )} .

It is then apparent that BP is finite since, with the notation

bk

=

ak,p-k,

we get

whence . 00

LJAn 1

furnishes the statement.

00

= LJBP ::._2

I

(1.1.23) If m E N and An is countable for each n E Nm, then well. Proof. Let An= A 1 for n EN, n ~m + 1; then

m

U An is countable as 1

(1.1.24) The set Q is countable. 00

Proof. Q =

U Qm where 1

Qn is countable according to (1.1.23), as well as Q according to (1.1.22).

I

(1.1.25) If A =I= 0 is countable, then the set of all.finite sequences in A is countable as well. Proof. Let Sm (m EN) be the set of m-sequences in A and A = {an: n EN}. If bn denotes the I-sequence of the unique member am then S1 = {bn: n EN}, hence S1 is countable. As we get an (m + !)-sequence by joining to an m-sequence an element of A as (m + l)th member, we obtain from every element of Sm a countable subset of Sm+i, hence the countability of Sm implies by (1.1.22) that 00

Sm+i is countable as well. Finally, again by (1.1.22),

USm is countable. I 1

24

1.1.

SETS

(1.1.26)

An important example of an uncountable set is the following: (1.1.26) For a, b ER, a < b, the interval (a, b) is uncountable. Proof. First we notice that if c < d, x E R, then there exists an interval [c', d'J c c (c, d) with c' < d', x ~ [c', d']. In fact, in the case x ~ (c, d), we can choose

c' = 2c + d' d' = c + 2d ; if x E (c, d), then c' = 2x + !! ' d' = ~ 3 3 3 is suitable. Suppose now (a, b) = {xn: n EN}. Let [ai, bi] c (a, b) be chosen a1 < b1 and x 1 ~ [a1 , b1 ], then [a 2, h2 ] c (ai, b1) so that a 2 < b2 , x 2 ~ and so on. Evidently a < a1 < a 2 < ... , b > b1 > b2 > ... , and for all n. Let x = sup {an: n EN}.

x +3 2d

so that [a 2 , b2], an < bn

Then an ~ x for each n, and since an < bm for each n and m, x ~ b m for each m, so that x E [am bn] (n EN). Hence x E (a, b), x i= Xn (n E N). Th1s is a contradiction. ll 1.1.h. Well-ordering. Let A be an ordered set with the ordering relation B2 holds for Bi, B 2 E m), then the union of the sets from mbelongs to 121: as well. A system of sets with this property is said to be inductive. Further, the set A E 121: is called maximal in 121: if A' = A whenever A' E 121:,A' ::> A. Thefollowing important theorem now holds: (1.1.28) Kuratowski-Zorn lemma. If 21: is an inductive system of sets and A 0 E 121:, then there exists a set A E 121:, maximal in 21:, for which A 0 c A.

The proofs of (l.1.27) and (1.1.28) (the latter in a somewhat stronger form) can be found e.g. in the following book: J. L. Kelley, General Topology. D. van Nostrand Company, 1955, p. 33.

1.1.h.

(1.1.28)

25

WELL-ORDERING

1.1.j. Exercises. 1. Prove on the pattern of (1.1.4)

(nA;) u ( nBj) = n{A; u B;: i E I, j E J} . iEI

jEJ

2. Show that A c B implies A - C

=A

n (B -

C).

3. Prove

U A; iEI

-

U B; c U (A;

iEJ

- B;).

iEI

4. Suppose that ~ is a reflexive and transitive relation for the elements of a set A (i.e. if x, y, z E A, then x ~ x, and x ~ y, y ~ z implies x ~ z) and let x "' y be valid iff x ~ y and y ~ x. Show that "' is an equivalence relation. 5. Let 0 # A; c R (i E I), A = U Ai be bounded above. Show that iEJ

sup A= sup {sup A;: i EI}. 6. Suppose that 0 # A c R is bounded, c ER, B = {c + a: a EA}. Prove that sup B = c

+ sup A,

inf B = c

+ inf A.

7. Suppose that 0 #-Ac R is; bounded, c ER, B = {ca: a EA}. Prove that sup B = c sup A, inf B = c inf A (c ~ 0), sup B = c inf A, inf B = c sup A (c ;;;; 0). 8. Show that the set of intervals on R having rational end points is co untable 9. The number x ER is said to be an algebraic number if there are an n EN and integers a0 , ••• , an such that (*)

Prove that the set of all real algebraic numbers is countable. [For given n and a0 , •• • , an, (*) is fulfilled by at most n distinct values of x.] 10. Prove that the system of finite subsets of a countable set is countable. 11. Prove that, for a, b E R, a < b, the set of the irrational numbers belonging to (a, b) is uncountable. 12. Prove that the numerical sequences (an) with the property an = 0 or an = I (n E N) constitute an uncountable set. 00 [Whenever O ;;;; x ;;;; I, there exists a sequence of this kind with x = L 2-nan-] 1 13. Let ~ be the system of all subsets of the set A. Prove that (a) if A is finite, then \ls is finite as well; (b) if A = N, then \ls is uncountable.

26

1.2.

EUCLIDEAN SPACES

(l.2.1)

[For Pc N let an = 1 or O according as n E Porn~ P.] 14. Let A be the set of the pairs (x, y) where x, y ER and let us agree that (x1, Y1) E A, (x2, Y2) E A implies (x1, yJ < (x2, yJ if either x1 < x 2 or x1 = x 2 and Yi < y 2 • Show that this is an ordering of the set A. 15. Consider Ac R with the relation < on Rand decide whether A is wellordered if (a) A= Q; (b) A= l; (c) A is the set of integers; (d) A= Nu {- ~: n E NJ;

EN} .

(e) A = {m - n : 1 : m, n 16. Suppose/that 2.( is an arbitrary and mc 2.( a disjoint family of sets. Show that there exists a disjoint system Wl such that mc Wl c 2.(, and if Wl c ~ c 2.( and ~ is disjoint, then ~ = Wl. 17. Show that there exists a set H c R with the property that every x ER has a unique representation in the form x

= .

n

L1 qihi, where n E N, q1 E Q, h1 E H

(i = 1, ... , n); by the uniqueness of the representation we understand that n

I1

n'

qih,

=

I1

q;h;

(q;, q1E Q, q1 =I= O =I= q1, h1, h1EH, h11 =I= h,. if i1 =I= i 2, h11 =I= hh if A =I= j 2) implies that n = n' and_ the hi are equal to the h except for order, finally, qi= q for hi= h;. [Consider the subsets F of R for which n E N, Ji, .. .,fn E F, /i1 =I= Ji, (i1 =I= iJ,

1

1

n

q1 E Q, }: qJi

= 0 imply q1 = . . . = qn = 0 and let H be a maximal set among

1

them.] 18. Construct a well-ordered uncountable set W such that the set W(x) = = {y: y < x} is countable for x E W. [Choose a well-ordered· uncountable set and the smallest element of those for which W(x) is uncountable.]

1.2. EUCLIDEAN SPACES 1.2.a. Distance. It is well known that, in a rectangular coordinate system, the points of the plane are characterized in a one-to-one manner by pairs (x1, xJ of real numbers, and the points of space by triples (xi, x 2 , x 3) of real numbers. The distance between the points (x1 , x 2) and (Yi, yJ of a plane is· then given by

Jcx1 - yi)2 + (x2 -

yJ 2

. 1.2.a.

(1.2.2)

27

DISTANCE

while the distance between the points (xi, x 2, x3) and (Yi, y 2, y 3) of the space is · given by J(x1 - yJ 2 +.(x2 - Y2) 2 + (xa - Ya) 2 .

It is known as well that in some cases, e.g. for the representation of the variables of a function of several variables, it is useful to consider m-sequences ' of real numbers as points of a "space" and to define the "distance" between two sequences of this type analogously by

Let us agree to understand by m-dimensional Euclidean space (denoted by Rm) the set of m-sequences (xi, ... , x;;;) of real numbers. Two sequences of this type are said to be equal iff their members agree, ordering included, i.e. (xi, ... , xm) = (Yi, ... , Ym) iff xi= Yi (i = 1, ... , m). The m-sequences themselves, i.e. the elements of Rm, are called the points of the m-dimensional Euclidean space, the numbers xi, ... , Xm are called the coordinates of the point x = = (xi, ... , Xm). The distance between the points x = (xi, ... , Xm) and y = = (Yi, .. , , Ym) is, by definition (1.2.1)

p(x, y)

=

J(x1

-

yJ 2

+ ... + (xm

- ym) 2 •

According to our agreement, a plane can be identified with two-dimensional space R2, and ordinary space with three-dimensional space R3 . Our definitions have a meaning even when m = 1, arid if the I-sequence (x) with its unique member x is identified with the real number x, the one-dimensional space R1 can be identified with the set R of real numbers, i.e. with the real line; this agrees · with the fact that, when m = 1, p(x, y)

= J(x - y)2 =Ix - YI

furnishes the distance between points with coordinates x and y on the real line. The above terminology is justified by the fact that the distance defined by (1.2.1) has several. essential properties generalizing basic properties of the distance between points on a plane and in space. This is expressed by the following theorem: (L2.2) For arbitrary points x, y, z E Rm, (a) 0 ~ p(x, y) < + oo, p(x, x) .= O; (b) p(x, y) = p(y, x); (c) p(x, z) ~· p(x, y)

+ p(y, z);

(d) p(x, y) = 0 implies x = y.

28

f.2.

(1.2.3)

EUCLIDEAN SPACES

Proof. (a), (b), and (d) are immediately obtained from (1.2.1). In order to prove (c), let

The inequality to be proved is

Ji

(x; - Z;)2

~

Ji

(xi - Y;)2

J~

+

(Y; -

Z;)2 •

It can be written

in which a;

=

b; =

Y;,

X; -

Y; -

Z; •

It is sufficient to prove instead the inequality obtained by squaring both sides: m

I· 1

m

(a; + b;) 2 =

I1

m

ar + 2 I a;b; + 1

m

I1

b; ~

which is equivalent to the Cauchy inequality (1.2.3) In order to prove (1.2.3), consider the polynomial of second degree m

P(u) = L (a;u 1

+ b;) 2 =

m

m

u 2 Lal+ 2u L a;b; 1

1

m

+ L bl;?;;

0.

1

Since P(u) ;?;; 0 the discriminant of the polynomial cannot be positive, hence

From this, (1.2.3) follows immediately. I When m = 2 or 3 and x, y, z are the vertices of a triangle, inequality (c) becomes the well-known inequality between the sides of a triangle, therefore it is called the triangle inequality. Observe that the following definitions are all based on the notion (1.2.1) of distance, and that, in a considerable number of the theorems based on them,

(1.2.6)

1.2.b.

29

CONVERGENCE OF A SEQUENCE OF POINTS

only the properties (l.2.2) of the distance are used; moreover, in most of them, (1.2.2) (d) is superfluous. The latter theorems will be marked by a star, while those which are based only on the properties (1.2.2) including (1.2.2) (d), will be marked by two stars. 1.2.b. Convergence of a sequence of points. We say that 'the sequence (an) of real numbers converges to the real limit a if, for each 8 > 0, there is an index n0 such that I an - a I < 8 whenever n ~ no, More intuitively, the set on the real line of the x fulfilling the inequality I x - a I < 8, i.e. the interval (a - 8, a + 8), may· be called the neighbourhood with radius 8 around the point a. Then an -+ a iff each neighbourhood of the point a contains all members of the sequence from a suitable index onwards. ' Let us accordingly consider in Rm the set of points x ERm having a distance smaller than e > 0 from a given point a ERm: S(a, 8}

= {x: p(x,a) < 8}.

This set is identical, when m = 2, with the circular disk of centre a and radius 8 (without the circumference) and, if m = 3, with the sphere of centre a and radius 8 (without the spherical boundary surface). We agree to call the set S(a, 8) Rm the ball with centre a and radius 8, or the 8-neighbourhood of the point a E Rm. ) It is now evident how to define the c9nvergence of a sequence (xn) consisting of points of the space Rm: we say that the sequence (xn) of points converges to the limit point y if, to each neighbourhood S(y, 8) of the point y, there e~.ists an index no such that Xn ES(y, 8) whenever n ~ n 0. In other words, this is true if, for every 8 > 0, there exists an n0 with p(xm y) < 8 if n ~ n0; in terms of the notion of the limit of numerical sequences, this means

c:

I

We denote by the symbol Xn -+

y . or lim Xn. =

yJ

the fact that the sequence (xn) of points belonging to Rm converges to the limit point y E Rm. · · · Since the above definition of the limit of a sequence of points can be based on the limit of numerical sequences [as it requires the convergence to O of the numerical sequence (p(xm y))], it follows easily in view of the basic properties of numerical sequences that ·the following theorems hold: (1.2.4)* The sequence of points x, x, x, . . . converges to· x. (1.2.5)* If xn

-+

I

y, then each subsequence of (xn) converges to y as well.

(1.2.6)* If X,; -+ z and Yn-+ z, then the sequence Xi, Yi, x 2, y 2, also converges to t, I

••• ,

I

Xm Ym ...

1.2.

30

(1.2.7)

EUCLIDEAN SPACES

(1.2. 7)* The convergence is not influenced by inserting or changing a finite number of members in a sequence. • . ' (1.2.8)** A sequence of points can possess one limit at most. Proof. If Xn -+ y and Xn -+ ·z, then by the triangle inequality 0;;:; p(y, z);:;:; p(y, Xn)·+ p(xn, z)

using (1.2.2) (a) and (b) too. Thus p(y, z) (d), y = z is true. I

= p(xn, y) + p(xm z)-+

0,

= 0, and, as a consequence of (1.2.2)

The following theorem states that the determination of the limit of a sequence of points can be reduced to that of the sequences of coordinates: (1.2.9) Xn -+ y ERm iff, using the notations Xn = (Xn1, ... , Xnm), y Ym), (i = 1, ... , m). lim Xni = Y, Proof. For a

(1.2.10)

=

I aj

(ai, . .. , am), b

= (b1 ,

- bj I ;;:; p(a, b) ;:;:;

m

... ,

L1 I ai -

= (yl, ... ,

bm) we easily get from (1.2.1) bi I

(j = 1, ... , m),

since

Hence 0 ;;:; I Xnj - Yi

I ;;:;

m

p(Xn, Y) ;:;:;

and this implies the statement.

L I Xni i=l

Yi

I

(j = 1, ... , m),

I

We say, by extending the terminology used for numerical sequences, that the sequence (xn) of points in Rm is convergent if it has a limit. A condition, similar to the Cauchy criterion for numerical sequences, can be given here for the convergence. In order to formulate this briefly, let us call a sequence (xn) of points in Rm a Cauchy sequence if, to every e > 0, there is an index n0 such that p(xn, xm) < e whenever n, m ~ n0 • (1.2.11)* A convergent sequence (xn) is a Cauchy sequence. Proof. If Xn -+ y, then, for e > 0, there is an n0 with p(xm y)

< ; for n ~ no.

Thus, according to the triangle inequality and by (l.2.2) (b), n, m

~

n0 implies

1.2.b.

( 1.2.15)

31

CONVERGENCE OF A SEQUENCE OF POINTS

(l.2.12) If (x,J is a Cauchy sequence in Rm, then it is convergent. Proof. For a given s > 0, let n0 be an index such that p(xn, x,;,) < e if n, m ~ n0 • Hence, with the notation ( 1.2.10) implies (n, m

~

n0 ,

i = 1, .•. , m).

This shows that each sequence (xn;) (i = 1, ... , m) of coordinates satisfies the Cauchy condition and converges by (1.1.15) to a limit Y;: limxni

Hence, according to (1.2.9),

= Yi

Xn ---+

(i

= 1, ... , m).

y for the point y = (Yi, ... , Ym) E Rm.

I

In order to extend the Bolzano-Weierstrass theorem from numerical sequences to sequences of points, we have to define first the notion of bounded sequences of points and, more generally, the notion of bounded point sets. The set 0 ¥= A c Rm is said to be bounded if the set

{p(x, y) : x, y EA} c R is bounded; in this case, the least upper bound of the latter set is called the diameter of A and is denoted by o(A): (1.2.13)

c>(A)

= sup {p(x, y) : x, y E Aj.

We agree. to consider the empty set bounded and define its diameter as equal to zero: o(0)

=

o.

This definition furnishes, in the case m = 1, sets of numbers bounded in the usual sense. Moreover, o(A) agrees, in the case of a circle or a sphere, with the elementary geometrical meanirig of the diameter. The definition immediately implies: (1.2.14)* Every subset of a bounded set is bounded, and o(A)

~

o(B) whenever

AcB.I

(1.2.15)* The union of a finite number of bounded sets is bounded, and a EA, b EB implies o(A u B) ~ o(A) + o(B) + p(a, b). Proof. From x EA, y EB we obtain, by using the triangle inequality twice,

p(x, y) ~ p(x, a)

+

p(a, b)

+

p(b, y) ~ o(A)

+

p(a, b)

+

o(B).

II

1.2.

32

(l.2.16)

EUCLIDEAN SPACES

(1.2. I 6) 0 =/= A c Rm is bounded (ff the set of all coordinates of the points of A is bounded. · Proof. I.,et a EA be fixed, x EA arbitrary, a = (ai, ... , am), x = (xi, ... , Xm)• Whence; by (1.2.10),

= I, ... , m).

(i

Thus, if A is bounded, then p(x, a)

~

t5(A) implies

for every x EA and for each i. On the other hand, if C~X;~d

for each i and x = (xi, ... , xm) EA, then x, y EA implies by (1.2.10)"" p(x, y)

~

m(d - c),

t5(A)

~

m(d - c).

I

The sequence (xn) of points in Rm is bounded if the point set

is bounded. Then the following theorem holds: (1.2.17) Bolzano-Weierstrass theorem. From every bounded sequence (xn) of points in. Rm, a convergent subsequence can be selected. Proof. By (1.2.16), the set of all coordinates of the points

is b.ounded. Hence, according to the Bolzano-Weierstrass theorem valid for numerical sequences, the sequence

of the first coordinates has a convergent subsequence, i.e. there exists a strictly increasing sequence (n!) (k EN) of natural numbers such that, for a suitable Yi ER, lini Xnp = YI. k-oo

Then, from the bounded sequence (xni, 2) of coordinates, a convergent subsequence can be selected, i.e. a subsequence (nf) of (n!) can be given with the property IimXnp = Y2• k-oo

(l.2.20)

1.2.C.

33

OPEN AND CLOSED SETS

In general, if the sequence (nD (k E N) is already chosen, let (nt+ 1) (k EN) be a subsequence such that lim Xnt+',i+l =

Yi+l •

k-oo

Finally the sequence (n'l:) (k EN) will be a subsequence of all sequences (nD (k EN) (i = 1,.. , m), thus lim

Xn;;',i

k-oo

(i = 1, .. , m).

= Yi

By (1.2.9), this means that lim Xn;;' = Y = (yi, ... , Ym) E Rm. k-oo

I

1.2.c. Open and closed sets. The spherical neighbourhoods used in the definition of the convergence of sequences of points give us a method to characterize the position of a point x E Rm relative to a set A c Rm. The point xis said to be an interior point of the set A if there exists a neighbourhood S(x, e) of x which is a subset of A : S(x, e) c A. The point x is said to be an exterior point :,f A if there exists a neighbourhood S(x, e) of x which does not intersect ,I!_: S(x, e) n A = 0. Finally xis a boundary point of A if every neighbourhood S(x, e) of x intersects A but none of them is a subset of A:

S(x, e) n A

=f,

0

=f,

S(x, e) - A.

From the definition it follows that (1.2.18)* For any set AC Rm and any point x E Rm, exactly one of the following possibilities is fulfilled: (a) x is an interior point of A; (b) x is an exterior point of A; (c) x is a boundary point of A. I (1.2.19)* Let Ac Rm and B = Rm - A be the complement of the set A. Then (a) the exterior points of A are identical with the interior points of B; (b) the boundary points of A are identical with the boundary points of B. I

It follows from x E S(x, e): (1.2.20)* The interior points of A belong to A, while the exterior points of A do not belong to A. I For the boundary points such a general statement cannot be given. E.g. it i~ evident that, on the real line, the interior points of the interval [O, 1) (closed on the left) are identical with the points of the open interval (0, 1) and its exterior points are identical with the points of the set ( - oo, 0) u (1, + oo ). Of its two boundary points O and 1, the first belongs and the second does not belong to the interval [O, 1). 3 Akos Csaszar

1.2.

34

(1.2.21)

EUCLIDEAN SPACES

It is worthwhile, in particular, to consider those sets which fulfil in this regard one of the two extreme possibilities. A set is open in Rm if it does not contain any of its boundary points, and it is closed if it contains all its boundary points. Our terms are in· accordance with the usual terms of open and closed intervals of the real line since, for a, b ER, a < b, (a, b) is an open and [a, b] is a closed point set of R. Indeed, let us more generally designate respectively as an open brick (ai, ... , am; bi, ... , bm) and a closed brick [ai, ... , am; bi, ... , bm] in Rm, for a;, b; ER, a; < b; (i = I, ... , m), those sets of points x = (x1, . • . , xm) E Rm for which a; < X; < b; and a; ~ X; ~ b; (i = I, ... , m). The term "brick" is motivated by the case m = 3. Then the following statement is valid: (1.2.21) The open bricks are open, the closed bricks are closed point sets in Rm. Proof. If a; < x; < b; for each i, and a > 0 is smaller than any of the numbers X; - a; and b; - X; (i = I, ... , m), then

for x = (xi, ... , xm) since y = (Yi, ... , Ym) E S(x, a) implies, in view of (1.2.10), a; < Y; < b; for each i. Hence every point of (a1 , . . . , am; bi, ... , bm) is an interior point; it does not contain any of its boundary points. On the other hand, if x ~ [ai, ... , am; bi, ...• bm], then xi < ai or xi> bi for at least one j. If a> 0 is smaller than (ai - xi) ot (xi - b), then, for any point y E S(x, a), Yi < ai or Yi> 'bi respectively by (1.2.10), thus

and x is an exterior point of [ai, ... , am; bi, ... , bm]. Therefore a closed brick contains all its boundary points. I Further examples are given by the following theorem: (1.2.22)* For any a E Rm and e > 0, the ball S(a, e) is open and the set S(a, a)= {x: p(x, a)~ a} is a closed point set in Rm. Proof. If x E S(a, a), i.e. p(x, a) < e, let c5 equality, y E S(x, c5) implies p(a, y)

~

p(a, x)

+ p(x, y)
e, so that S(x, c5) n S(a, a) = 0

(1.2.27)

1.2.c.

35

OPEN AND CLOSED SETS

where b = p(x, a) - e because y E S(x, b) n S(a, e) would imply p(x, a) ~ p(x, y)

+ p(y, a)
0 smaller than each number 8;). Hence every point ofA is an interior point of A. I

We can easily get from (1.2.28) and (1.2.27), or from the definitions: (1.2.30)* The empty set and Rm are simultaneously open and closed. I 1.2.d. Dense sets. We know from (1.1.11) the property of the set of rational

numbers on the real line that every non-empty open interval contains a rational number; this means in other words that every neighbourhood S(x, 8) of each point x E R contains a rational point. This condition can be briefly stated as follows: the set of rational numbers is "dense" (or "everywhere dense") on the real line. According to this, the set Ac Rm is said to be dense (or everywhere dense), if every neighbourhood S(x, 8) of each point x E Rm intersects A. As weUas on the real line, there is a countable dense set in Rm too: (1.2.31) The set of points with rational coordinates is countable and dense in

Rm.

(1.2.33)

1.2.d.

37

DENSE SETS

Proof. The set A in question is countable since the set of sequences of m members in the countable set Q is countable ((1.1.24) and (1.1.25)). Let x E Rm and e > 0 be arbitrary, x = (xi, ... , Xm)- There is, for every i, a rational

Yi with IYi -

X;

I
0, the set of balls with centres belonging to A and radius e is countable. Suppose now that A c Rm is dense, 0 # G c Rm is open and x E G. For suitable e > 0, S(x, e) c G and, by hypothesis, there is an a EA such that

a E S (x,

~ ) . Let

6 be a rational number such that p(x, a) < 6
S. For any point x E A, we can find a set CE (£ for which x E C and, expressing this open set a; as the union of sets from >S, one of the members B E 5S contains x, i.e. x EB c C. Let us do this for each point x of A; the sets B E5S occurring here can be ordered in a sequence (B;) (i E N) because of the countability of >S, and, for every B;, there is a C; E (£ with B; c C;. Consequently every x E A belongs to a B; and so to a C;: the system {C;: i EN} is a countable covering of A selected from I

a:.

(1.2.35) The Borel covering theorem. From any open covering (£ of any bounded, closed set Kc Rm, a finite covering of K can be selected. Proof. According to (1.2.34), a countable covering {C;: i EN} of K can be selected from We assert that for large enough n, even {C;: i = l, ... , n} is a covering of K. Otherwise let n Kn = K - U C; # 0.

a:.

i=l

n

By (1.2.28),

U C; is

n

open and, in view of (1.2.27), Rm -

i=l

U C;

is closed; hence

i=l

by (1.2.28) again,

n

LJ C;)

Kn = Kn (Rm -

i=l

is closed. Evidently

Thus the sets Km together with K, are bounded by (1.2.14). Therefore, on account of (1.2.33), oo n 00 Kn = Kn (Rm - U C;)

n 1

n

n=l

i=l

oo

= Kn (Rm -

n

U UC;) n=li=l 00

00

= Kn (Rm - U C;) = K - U C; # 0 1

1

in contradiction to the fact that {C;: i EN} is a covering of K.

I

1.2.f.

(1.2.35)

39

EXERCISES

1.2.f. Exercises. 1. Suppose x = (xi, . .. , Xm), y = (Yi, ... , Ym), = (y~, ... , y;,,), x, y, x', y' E Rm and m

x; =

I

x' = (x~, ... , x;,,),

y' =

m

I

auxj, y; =

j=l

(i

auyj

= 1, ... , m),

j=l

where (p, q = 1, ... , m),

and Jpq = 0 if p i= q, Jpq = 1 if p = q. Prove that p(x, y) = p(x', y'). 2. Prove that, if x, y, z E Rm, then

I p(x, y)

- p(y, z)

J(T) = 4. Determine Ac Rm if (a) A (b) A (c) A (d) A

J~

I~

p(x, z).

(a; - b;) 2 •

the interior, exterior, boundary, and limit points of the set

S(a, e), a ERm, e > O; S(a, e), a E Rm, e > O; (ai, .. ., am; b1 , •• . , bm), a;< b,; [a1, .. • , am; b1 , •• • , bm], a;< b;; (e) A = {(xi, ... , xm) : X; EN. (i = 1, ... , m)}; (f) A = {(xi, ... , Xm) : X; E Q (i = 1, ... , m)}. 5. Show that x E Rm is a limit point of the set Ac Rm iffthere exists a sequence (xn) in A with the property Xn - X. 6. Prove that, if Ac Rm is open (closed) and B c Rm is closed (open), then A - Bis open (closed). OCJ 7. Construct open (closed) sets An (n EN) in R such that An is not open

= = = =

n 1

OCJ

( U An is not closed). 1

8. Show that if each set A; c R is dense in R (i = I, ... , m), then the set A

= {(xi, ... , Xm):

X;

EA; (i = I, ... , m)}

is dense in Rm. 9. Prove that, if@ is a disjoint system of open sets in Rm, then @ is countable. 10. The point a E Rm is said to be a (strict) maximum point of the real function f on Rm if there exists an e > 0 such that f(x) ~ f(a) (f(x) < f(a)) holds for x E S(a, e), x i= a. Show that if A is the set of the maximum points and B is

1.3.

40

(1.3.1)

METRIC SPACES

the set of the strict maximum points of the function f, then B and the set {f(x) :xEA} are countable. [Let An and Bn be the sets of those points for which, in the definition of maxi-

mum (strict maximum), e = _!__ can be taken, and observe that, for x, y EBn, n x# y, as well asin the case x,yEA 11 ,f(x) f=f(y), we have S (x, 2~) n s(y,

;n) = 0.J

11. Let/be a real function on R with the property that, for each a ER, there exists an e > 0 such that f(x) ;£ f(a) for a - e < x < a and f(x) '?:. f(a) for a < x < a + e. Prove that f is monotone increasing. [Otherwise a sequence of intervals

could be constructed such that bn - an

--+

0 and f(an)

> f(bn).]

1.3. METRIC SPACES 1.3.a. Metric. Besides the convergence of numerical sequences and sequences of points in Rm, the convergence of sequences of functions plays an important role in analysis. Various types are known: e.g. a sequence of real functions In on a set H # 0 is said to converge uniformly to the limit function f on H if, for every e > 0, there is an index n 0 such that n '?:. n0 implies I fn(t) - f(t) I < e for each t E H. This type of convergence plays an important role in many investigations. We can also define other types of convergence, e.g. for real functions continuous in some interval [a, b] of the real line. The sequence (fn) of such functions is said to converge in the mean to the first or second power to the limit function f, continuous as well, if b

Jim

J lfn(t) -f(t) I dt = o,

n-oo a

or b

lim

J

I

f,,(t) - f(t)

12

dt

=o

n-oo a

respectively. For the purpose of examining mean convergence to the first power, it is practical to define the "distance" between the functions f and g of the set E of real functions continuous on [a, b] as follows: b

(1.3.1)

p(f, g)

=

Slf(t) -

a

g(t) I dt.

(1.3.4)

1.3.a.

41

METRIC

The convergence of the sequence (!,,) to the ·limit function Jin the given sense postulates then the fulfilmentof the condition p(J,.,f) ---t Ojust as the convergence of the sequence of points (xn) of Rm to the limit point x requires the condition p(xm x) ---+ 0, naturally for the distance (1.2.1) defined in Rm. That the expression (1.3.1) is called the distance between J and g is conditioned by the fact, easily proved, that it has the properties (1.2.2) (a)-(d); (c) follows from the inequality IJ(t) - h(t) I ~

(1.3.2)

I/Ct) -

g(t)

I + I g(t)

- h(t)

I

(a~ t ~ b)

and (d) from the fact that the integral of a non-negative continuous function can be O only if the function itself is identically 0. A similar consideration can be used in the case of uniform convergence, at least if we restrict ourselves to bounded functions on the underlying set H. Let us denote by E the set of these functions and put (1.3.3)

p(J, g)

= sup {I J(t)

- g(t)

I : t EH}

for J, g EE. It is evident that the sequence (J,.) from E converges uniformly to the limit function JEE iff p(f,,,J)---+ 0, and one can easily convince oneself that the properties (1.2.2) (a)-(d) hold again; the validity of (c) follows from (1.3.2) as t E H implies

I/Ct) -

h(t)

I~

+ p(g, h),

p(J,g)

hence p( f, h) ~ p(J, g)

+ p(g, h) ,l

In order to examine the mean convergence to the second power, it would be desirable again to denote by E the set of the continuous functions on [a, b] and to understand by the distance between f and g (J, g EE) the value b

Slf(t) a

g(t)

1

2dt.

However, in this c!l,se, the property (1.2.2) (c) would not hold in general. Nevertheless, if we accept the definition b

(1.3.4)

p(.f,g)

=

CS lf(t) -

g(t)

12a,)112,

a

we can still say that p(f,,,f)---t Ois the condition of the convergence now studied; (1.2.2) (a), (b), (d) and, moreover, (c) are valid. This can be proved as in the case of the distance defined in Rm: with the notation f - g = p, g - h =;= q, on account off - h = p + q, the inequality to be proved can be written in the form

cs

b

a

(p(t)

+ q(t))2dt)1'2

~

cs

b

a

p(t)2dt) 1' 2

+

cs

b

a

q(t)2dt)1' 2 ,

1.3.

42

i.e.

s b

a

~

s b

s b

p(t) 2dt + 2

a

s b

p(t) q(t) dt +

a

q(t) 2dt ~

a

b

p(t)2dt

(1.3.5)

METRIC SPACES

b

+ 2 (S p(t)2dt)112 (5 q(t)2dt)112 + a

a

s b

q(t)2dt,

a

consequently it can be reduced to the Schwarz inequality

s b

(l.3.5)

p(t) q(t) dt ~

a

b

b

(S p(t)2dt) 1' 2 (5 q(t)2dt)1' 2• a

·

a

In order to prove this, we have again to consider the discriminant of the polynomial b

P(u) =

b

b

b

J (up(t) + q(t)) dt = u Jp(t)2dt + 2u Jp(t) q(t)dt + J q(t)2dt ~ 0 2

2

a

a

a

a

and observe that this cannot be positive, so that we get (1.3.5). The examples presented here show that in many cases it is possible and useful to introduce a "distance" with the properties (1.2.2) (a)-(d) among the elements of some sets (e.g. sets of functions) other than Rm. The underlying set acquires in this way a geometric structure. This leads to the following definition: If, for arbitrary elements x, y of a set E =I= 0, a real number p(x, y) is defined with the properties (D 1) CD2) (D 3) (DJ

~ p(x, y) < + oo, p(x, x) = 0 (x, y EE); p(x, y) = p(y, x) (x, y EE); p(x, z) ~ p(x, y) + p(y, z) (x, y, z EE); p(x, y) = 0 impiies x = y,

0

then the set E is said to be a metric space, its elements are called points and p(x, y) is the distance between the points x and y. A real function p of this kind is a metric on E. Accordingly, the Euclidean space Rm (for any m) with the metric (1.2.1), the set of real functions bounded on a set H with the metric (1.3.3), the set of real functions continuous on [a, b] with the metric (1.3.1) or (1.3.4), are all examples of metric spaces. The two latter examples show that several metrics can be defined on the same set, and the metric space is determined only if the metric is exactly described; consequently it would be necessary to formulate the preceding definition of metric space more exactly. by saying that a metric space is a pair [E, p] consisting of a set E and a metric p defined on it. However, it is in general clear from the context which metric is to be taken on the set E, and it leads to no misunderstanding if we call the set E itself a metric space. We shall denote the metric (1.2.1) defined on the set Rm by Pm· As a further example, let E =I= 0 be an arbitrary set and, for x, y EE, define p

if (x, y ) -_ { .0, 1 if '

X

= y, 4

X,...

y.

(l.3.6)

1.3.b.

CONVERGENCE

43

It is easily shown that we get in this way a metric space; we call this type of space a discrete metric space.

We can get further metric spaces in the following :way. Let [E, p] be a metric space, 0 # E 0 c E and, for x, y E E0 ,

Po(X, y)

= p(x, y).

p 0 , defined on the elements of E0 , fulfils then conditions (D1)-(D4), hence [E0 , p 0 ]

is a metric space again. The space obtained in this way is called a subspace of the space [E, p]; we use the notation (1.3.6)

Po=

PI Eo.

E.g. let E be the set of functions bounded on the interval H = [a, b] c R, p the metric (1.3.3), E 0 the set of functions continuous on H; then the space [E0 , p I E 0 ] is a metric space (and this is a further example for a metric defined

on the set E0 besides those in (1.3.1) and (1.3.4), studied before). In the same way, an arbitrary set 0 # E c Rm can be considered as a metric space with the metric Pm IE, 1.3.b. Convergence. Open and closed sets. We emphasized at the end of section 1.2.a that a large part of the definitions and theorems in 1.2 make use only of the properties (1.2.2) (a)-(d) of the metric (1.2.1). Therefore these definitions can be extended and the theorems in question are valid even if we formulate them for the points and subsets of an arbitrary metric space instead of Rm. In this way, for instance, the definition of the convergence of sequences of points, given in 1.2.b and relying on the balls S(a, e), can be applied to sequences of points of an arbitrary metric space, and the relation Xn - y is equivalent in this case again to the condition p(xm y) - 0. The various notions of distance introduced on some sets of functions in the examples 1.3.a lead us precisely in this way to the notion of uniform and mean convergence. The notion of Cauchy sequence, bounded set, diameter, bounded sequence, interior, exterior, and boundary point, open and closed set, closed ball, limit point, dense set, (finite, countable, open, closed) covering can be similarly extended to an arbitrary metric space and the theorems relating to them, marked by one or two stars in 1.2, are still valid (Rm must then be replaced by an arbitrary metric space). We emphasized also that in a considerable part of the theorems mentioned above (i.e. those marked by one star), we did not use the property (1.2.2) (d); in extending them to metric spaces, (D4) may therefore be omitted. We shall again mark by one star the statements in which we do not rely on (D4). In the theorems not marked with a star, on the other hand, we used not only the properties in (1.2.2) of the distance but directly the definition (1.2.1). Some of them (in which we refer to the coordinates of points in Rrn) cannot even be formulated in an arbitrary metric space; some others may be fdrmulated, but

44

1.3.

(1.3. 7)

METRIC SPACES

will not necessarily be valid in an arbitrary metric space. Those metric spaces in which some of these theorems are nevertheless true will be of great value of course, since they stand from a certain point of view nearer than general metric spaces to Euclidean spaces. These considerations lead us to the following definitions. 1.3.c. Complete metric spaces. According to (1.2.11), an arbitrary convergent sequence in an arbitrary metric space is a Cauchy sequence. The converse (1.2.12) of this is not true in every metric space. E.g. if E denotes the set of all rational points of the real line with the metric p 1 I E (i.e. E is considered as a subspace of R) and (xn) is a numerical sequence consisting of rational numbers and converging to an irrational limit, then (xn) is a Cauchy sequence not convergent in E. More generally, if a sequence (xn) converges to a limit point x in a metric space E and no member of the sequence is equal to x, then, in every subspace £ 0 of E containing all members of (xn) but not containing x, the sequence (xn) is a Cauchy sequence without being convergent (in fact, in view of (1.2.8), it cannot converge to a point belonging to £ 0). On the other hand, if E denotes the space of bounded functions on the set H # 0 with the metric (l.3.3) and (!,,) is a Cauchy sequence in it, then, for any fixed t EH, the numerical sequence (fn(t)) satisfies the Cauchy condition since, for e > 0, there is an n0 such that n, m ~ n0 implies p(fn,fm) < e and then, for the same indices n, m, If,,(t) - fm(t) I < e holds as well. Thus the sequence (f,,(t)) converges to a finite limit (in general depending on t); let us denote it by f(t). According to the preceding, for e > 0, there is an n0 such that n, m ~ n0 and t E H imply 11,,(t) - fm(t) I < e, hence lf(t) - fm(t) I ~ e (m ~ n0) for n - oo. This shows firstly that f is a bounded function, thus f EE; on the other hand p(f,fm) ~ e for m ~ n0 i.e. p(fmJ)-+ 0 if m -+ oo. Hence in this space all Cauchy sequences converge. Let us agree to call the metric space E complete if all Cauchy sequences are convergent in it. Hence we can say: (1.3.7) The metric space [Rm, Pm] is complete.

I

(1.3.8) If E is the set of bounded real functions defined on the set H # 0 and p is the metric (1.3.3), then [E, p] is a complete metric space. I The term "complete" is chosen because of the examples of non-complete spaces above in which we "left out" the limit points of some sequences from the space and obtained in this way "incomplete" spaces. Later results will illuminate this term still better. (1.3.9)* A closed subspace of a complete metric space is complete.

(1.3.10)

1.3.C. COMPLETE METRIC SPACES

45

Proof. If 0 # E0 c E is closed and (xn) is a Cauchy sequence in E0 , then it is a Cauchy sequence as a sequence in E, thus it converges to a point x of E. By (1.2.26), x EE0 , hence the sequence converges already in E0 • I

E.g., if 0 # H c Rm, and E is the set of bounded functions on the set H with the metric (1.3.3), and E0 denotes the set of all bounded, continuous functions on H, then E0 is according to (1.2.26) a closed subset of E since the limit function of a uniformly convergent sequence of continuous functions is continuous. Thus the space E 0 is complete as well. Conversely: (1.3.10) If E 0 is a complete subspace of the metric space E, then it is closed in E. Proof. In view of (1.2.26), it suffices to show that if Xn EE0 (n EN) and Xn --+ --+ y EE, then y EE0 • However, the sequence (xn), convergent in E, is a Cauchy sequence, thus, by the completeness of E0 , it converges to a point z EE0 • Now (1.2.8) implies y = z, hence y EE0 • I

1.3.d. Separable metric spaces. An important property of the space Rm is that (by (1.2.31)) there is a countable dense set in it. This is not true of every metric space. E.g. in a discrete metric space no proper subset can be dense, thus, if the underlying set is uncountable, there cannot exist a countable dense subset of the space. As a less trivial example, let Ebe the set of functions bounded on the interval [a, bJ c R with the metric (1.3.3). Consider, for each x E [a, b ], the function Ux whicltis equal to, 1 at x and to Oeverywhere else. It is obvious that p(gx, gy) = 1 for x # y. If now the countable set {f,, : n EN} c E were dense in E, then we could find, for every x E[a, b ], an index nx with the property p(gx,fn) < nx # ny whenever x # y, as nx

~ , and

= ny would imply

In view of this, we could order all numbers of the interval [a, b], according to the nx belonging to them, in a sequence, which is impossible by (1.1.26) and (1.1.19). On the other hand, let us consider the subset E 0 of the former space E composed of the functions continuous on [a, b ]. In this case, there exists already a countable dense set. We can get such a set A by considering all functions constructed in the following way: we divide the interval [a, b] into n equal parts (n = 1, 2, ...), we prescribe rational function values at the subdivision points and at the endpoints, and define the function to be linear on each subinterval belonging to the subdivision. In fact, it is clear that, for a given n, there exist countably many functions of this kind (countably many sequences consisting of (n + 1) members can be formed of the set of rational numbers), hence A is countable. Now let fEE0 and e > 0 be arbitrary, and, on account of the uniform continuity off,

46

1.3.

(1.3.11)

METRIC SPACES

. let n be so large that the oscillation off is smaller than ; on the subintervals of b-a

b-a

n

n

length - - of [a, b], then with the notation ti= a+ i - - ( i = 0, 1, ... , n), let Yi be a rational number satisfying the inequality

finally let g EA be a piecewise linear function taking the value y 1 at ti. Now if t E [a, b], then t;_ 1 ~ t ~ ti for some i, so that · lf(t) - g(t) I ~ lf(t) - f(t,)

I+

lf(ti) - g(t;)

I + Ig(t1)

- g(t)

I;

since

+ 1/Cti-J -

g(t;-1) I
0)

where tis an arbitrary element of Hand

8

denotes an arbitrary positive number.

54

2,1, THE NOTION OF TOPOLOGICAL SPACE

(2.1.1)

The fact that fn --+ f pointwise means in fact that, to every neighbourhood V,,. off, we can find an i'l0 such that f,, E V,,. for n ~. n0 • · Starting from these examples, the following general definition of convergence can be obtained: we assign to each element x of E some subsets of E, called "neighbourhoods of x", and say that a sequence (xn) consisting of elements of E "converges to x" if, for each neighbourhood of x, there exists an index no such. that xn E V for n ~ no. The only question is what conditions are to be assumed with respect to the sets designated as neighbourhoods in order to get a general notion of cbnvergence which includes the former examples and is useful. In this connection, let us first notice that it is quite natural to require that theorem (1.2.4) be fulfilled, i.e. that the sequence x, x, x, ... converges to x. This is ensured by the fact that each neighbourhood of x contains x itself; this was true in all our former examples and we shall assume it henceforth. On the other hand, we have to consider that the neighbourhoods of the point x may be defined in several different ways, so that the same notion of convergence arises from different notions of neighbourhoods. Thus it is firstly evident that, if the systems of sets ~ 1 and ~ 2 are related to each other in such a way that each set Vi E ~ 1 contains a subset V1 ::::, V2 E ~ 2 and each set of ~ 2 contains a subset from ~ i , then the same sequence will converge to x if we consider as its neighbourhoods the sets from ~ 1 or the sets from ~ 2 • We arrive e.g. at the usual notion of convergence on the real line if we consider as neighbourhoods of x, not the sets (x - s, x + 8) (8 > O) as usual, but the sets of one of the following systems of sets: (a) the sets [x - 8, x (b) the sets (x - 8, x (c)thesets

+ 8] + 8)

(x-!,x+

!)

(8

> O);

(8

> 0, rational);

(nEN);

(d) all open intervals containing x; (e) all open intervals with rational end-points containing x.

It would be easy to continue this list. We get in the same way, in a pseudo-metric space [E, p ], the usual notion of convergence by considering as neighbourhoods of x, instead of the balls S(x, 8) (8 > 0), one of the following sets:

> O);

(f) the closed balls

S(x, 8) (8

(g) the balls S(x, 8)

(8 > 0, rational);

(h) the balls S

(x, !)(n EN);

(i) all balls containing x;

(2.1.1)

2.1.b.

55

CENTRED SYSTEMS, FIL1ER BASES, FILTERS

(j) all balls with rational radii containing x; etc.

We can add to these, in the space Rm, e.g. (k) the open bricks (x1 i = 1, ... , m);

-

s1 ,

... ,

Xm - sm; x1

+

Bi, ... , Xm

+ Bm)

(I) the closed bricks [x1

-

Bi, • . . ,

xm - sm; x1

+

s1 ,

+

•.. ,

Xm

(s; > 0,

Bm] (s; > O);

(m) all open bricks containing x. Let us then notice that if we define the sequences converging to x by considering as neighbourhoods of x the sets belonging to a system of sets ~, and we add to the neighbourhoods of x the sets of the form V1 n V2 , where Vi, V 2 E ~' then the same sequences will converge to x. In fact, if xn E Vi for n ~ n1 and Xn E V2 whenever n ~ n 2 , then, if n0 = max(n 1 , n 2 ), Xn E Vin V2 is true for n ~ n0 • In the same way, it has no effect on the sequences converging to x if we add to the neighbourhoods of x the sets obtained as the intersection of a finite number of sets from ~Consequently two systems of neighbourhoods ~ 1 and ~ 2 lead to the same notion of convergence if each set from ~ 1 has a subset from ~ 2 and each set from ~ 2 has a subset from ~ 1 , as well as when ~ 2 consists of the finite intersections of sets belonging to ~ 1 . In view of the notion of convergence it makes no sense to distinguish between systems of neighbourhoods which are in a relation of the above type to each other. Before formulating the definitions, it is therefore useful to study more thoroughly the relation of this type among systems of sets and to introduce a suitable terminology. This is the purpose of the following section. 2.1.b. Centred systems, filter bases, filters. In what follows, we understand by a system of sets a non-empty system of sets. We shall frequently speak of a system of sets whose elements are subsets of a given set E; we speak then of a system of subsets of E or a system of sets in E. Let ill: and )S be systems of sets. We say that ill: is coarser than )S, or l8 is finer than ill:, denoted by ill: < )S or )8 > ill:, if each set A E ill: has a subset B c A, BE )8. We get immediately from the definition: (2.1.1) (a) If fil C )8 then fil < )8; (b) If m< m, m < (£ then

m
RE r, thus A =I= 0 by R =I= ff. If A1 , A 2 E Ill, there are R 1 , R 2 E r such that A1 :::::> R1 , A 2 :::::> R 2 , then R 3 Er such that R1 n R 2 :::::> R 3 , and finally A 3 E Ill such that R3 :::::> A 3 • Thus A1 n A 2 =:) A3. I

Accordingly, a system of sets equivalent to a filter in E must be a filter base. Conversely:

(2.1.9)

2.1.b.

57

CENTRED SYSTEMS, FILTER BASES, FILTERS

(2.1.6) If t is a filter base in E, then the system & ascending in E, generated by t, is a filter in E. It is called the filter in E generated by t, and t is called a base of the filter fr. Proof. If SE&, there exists RE t such that S:) R, thus R =I= 0 implies S =I= 0. On the other hand, if S 1 , S 2 E &, choose R 1 , R 2 E t such that S1 :J R 1 , S 2 :J R"' and R 3 E t such that R 1 n R 2 ::::> R 3 • Then S 1 n S2 ::::i R3 and S1 n S 2 E fr. I

E.g. the filter base consisting of the single element {A}, where 0 =/= A c E,. generates as a filter in Ethe principal filter A, and the filter base {{x}} consisting: of the single set {x} where x EE generates the fundamental filter x. In view of the preceding theorem, a base of a filter fr in E is a filter base t in E which generates fr as an ascending system in E, i.e. which is equivalent to fr. According to (2.1.2) and (2.1.5) we can say: (2.1. 7) A system of sets is the base of a filter 5 in E if! t c & and each set SE ~ has a subset R E t. I The systems of sets coarser than a filter are also of interest; the centred systems. are of this type. Let us call the system of sets c a centred system if a finite intersection of sets from c is never empty. Evidently any filter base, in particular any filter, is a centred system. (2.1.8) A system of sets coarser than a centred system is again centred. Proof. Let c be a centred system and \U: < c. Then for A; E \U: (i = I, ... , n), n

there are sets C; E c (i = I, ... , n) such that A;::::> C;;

1

n

n1 A; =I= 0.

n C; =/= 0 implies.

•

Thus a system of sets coarser than a filter base in E has to be centred. Conversely: (2.1.9) Let c be a centred system in E. The sets obtained as finite intersections· of sets from c then form a filter base in E. The filter 5 generated in E by t is finer than c and at the same time the coarsest (that is the smallest) filter in E which is· finer than c. We say that fr is the filter generated in E by c, c is a subbase of the filter fr. Proof. As c is centred, the sets belonging to t are not empty and the intersec-tion of two sets belonging to t itself belongs to r. Thus t is a filter base. Evidently c c t so that c < t < fr. If, for a filter fr' in E, c < fr' then c c fr' in view or· (2.1.2), hence t c fr', t < fr', and fr< t b(x) and t > > o(x) hold simultaneously. Consequently the notion of convergence defined in metric spaces is a special case of that defined in a neighbourhood space. We call the filter base assigned to the sequence (xn) by the formulae (2.1.26) and (2.1.27) the sequential filter base belonging to (xn). From the preceding definition, it follows easily that the fundamental properties of convergence (see (1.2.4)-(1.2.7)) still hold. The definition of interior, exterior, boundary and limit points of point sets can be generalized to neighbourhood spaces: we say that the point x EE is an interior point of the set A c E if there exists a neighbourhood VE b(x) of x such that V c A; xis an exterior point of A if there is a VE b(x) such that V n A = 0; x is said to be a boundary point of A if V n A #- 0 #- V - A for each neighbourhood VE b(x) of x. Finally x is a limit point of A if V n A #- 0 for each set VE b(x). Let us notice that xis an interior point of A iff b(x) > {A}, xis an exterior point of A ifftJ(x) > {E -. A}, xis a limit point of A iff 0E b(x) (n) {A}, xis a boundary point of A iff 0 Eb(x) (n) {A} and 0 Eb(x) (n) {E - A}. Hence it is evident that tJ(x) can be replaced by any other neighbourhood base o(x) of x in the definition, so that the corresponding notions in the case of metric spaces are special cases of those just defined.

62

2.1.

THE NOTION OF TOPOLOGICAL SPACE

(2.1.28)

The following theorem, corresponding to the theorems (l.2.18)-(1.2.20) and (1.2.24), immediately follows from the definitions: (2.1.28) Let [E, "\9] be a neighbourhood space, A c E, B = E - A, x EE. Exactly one of the following statements holds:

(a) x is an interior point of A; (b) x is an exterior point of A; (c) x is a boundary point of A.

x is an exterior point of A if! it is an interior point of B, and x is a boundary point of A if! it is a boundary point of B. If x is an interior point of A, then x EA. x is a limit point of A if! it is an interior or boundary point of it. I Evidently the definition of an open or closed set can be carried over too: we say that the set A c E is open in the neighbourhood space [E, "\9] if it does not contain any of its boundary points, and it is closed if it contains all its boundary points. From (2.1.18), we get the statements corresponding to (1.2.23) and (1.2.25): (2.1.29) The set A is open in a neighb~urhood space if! all its points are interior points and it is closed if! it contains all its limit points. I If we give the role of the balls S(x, a) to the neighbourhoods VE b(x), the proof of the theorems (1.2.27)-(1.2.30) can be repeated: (2.1.30) In a neighbourhood space [E, "\9],

(a) the complement of an open set is closed, the complement of a closed set is open; (b) 0 and E are both open and closed; (c) the union of open sets is open, the intersection of closed sets is closed; (d) the finite intersection of open sets is open, the finite union of closed sets is closed. Proof. (a) can be obtained immediately from (2.1.18) and (b) from the definitions. By (a) it is sufficient to prove the part relating to open sets of (c) and (d). If A; (i E I) is open and x E A = U A;, then x EA; for an index i0 EI; thus 0

iEl

there exists a VE b(x) with V c A;, c A, so that all points of A are interior points. n If A; is open (i = I, ... 'n) and XE B = A;, then there is a V; E b(x) such

n 1

that V; c A; and from the fact that b(x) is a filter, it follows: n

n

1

1

n V; E b(x), n V; c

B.

1

2.1.e. Topological spaces. It would be easy to go on adapting the notions and theorems known in metric spaces to arbitrary neighbourhood spaces. However, we shall not do that, for the notion of neighbourhood space is too general

(2.1.30)

2.1.e.

63

TOPOLOGICAL SPACES

for our purpose: we postulated in its definition too little from the neighbourhood filters assigned to the points so that some important statements which are valid in metric spaces no longer hold in arbitrary neighbourhood spaces. E.g. let E = R and ""§! be the neighbourhood structure on E where the system of sets {Vx} co~sisting of the unique interval Vx = (x - I, x + I) is a neighbourhood base of the point x. The theorem corresponding to (1.2.8) does not hold in the neighbourhood space [E, ~] since a sequence (xn) may converge to more than one limit point; e.g. the sequence 0, 0, 0, ... converges besides O to every point x with the property -1 < x < I. The fact that each point has a neighbourhood base consisting of open sets [valid in metric spaces in view of (1.2.22)] also fails to hold now. In fact, in this space 0 and E are the only open sets: if A is open and x EA, then (x -1, x + 1) c A, thus e.g. x +~EA and therefore (x - ~, x

+ ~) c

A, hence x

+ IE A and we see finally ttat

(x, +oo) C

c A and similarly (- oo, x) c A. Consequently, if we wish to come nearer to the relations valid in metric spaces, we have to make further restrictions on the neighbourhood filters of the points of the neighbourhood space. The first restriction of this kind is in connection with the circumstance studied just now, namely with the existence of neighbourhood bases consisting of open sets, and leads us to the following fundamental definition: The neighbourhood structure SJ given on the set Eis called a topology if every point x EE has a neighbourhood base consisting of open sets. If SJ is a topology, then the neighbourhood space [E, ST] is called a topological space. lf from the circumstances it is evident with which topology the set E is equipped, then the set E itself, instead of [E, ST], can be called a topological space. On the other hand, if several topologies are studied at the same time on the same set, we must take care to indicate to which topology the different notions introduced in topological spaces refer. Accordingly, we shall speak then of sets "open with respect to Sf" or briefly "SJ-open" etc. We can see from (1.2.22) that the neighbourhood structure defined in metric spaces satisfies this restriction, so that it is a topologyr, therefore we shall speak from now on of the topology of a metric space. We denote by STP the topology induced by the metric p. In particular, the topology on the real line belonging to the metric p 1(x, y) = Ix - y I will be denoted by $ and the one belonging to the metric (1.2.1) in the space Rm by $m; it will be called (one- or m-dimensional) Euclidean topology. Let us observe that if p is not a metric but only a pseudo-metric on the set E, it is still true that the balls S(x, a)= {y EE: p(x, y) < a}

(x EE,

B

> 0)

constitute a filter base consisting of sets which contain x so that, for a given x E E and e > 0, the family of these balls can be considered as a neighbourhood

64

2.1.

THE NOTION OF TOPOLOGICAL SPACE

base for a neighbourhood structure 8fP; S(x, s) will be open with respect to this as the proof of (1.2.22) will also be valid now. Accordingly if [E, p] is a pseudometric space, the topology 8!P will be called the topology of the pseudo-metric space. We can prove easily that the neighbourhood structure introduced above on the set Ru {- m, + oo} in order to define infinite limits is a topology too. Similarly, the neighbourhood structure defined on R in order to discuss the limit on the right is also a topology, since the sets [x, x+ s) constituting the neighbourhood base of the point x are open with respect to it (if x :;;; y < x + s then [y, x + s) c c [x, x + s)); we shall denote this topology by @+- and the topology introduced in an analogous way in order to study the limit on the left by &c-. The neighbourhood structure defined in order to study the pointwise convergence on the set E of the functions f defined on H, with the help of the neighbourhood sub base consisting, for f EE, of the sets

V1,.Cf) = {g: I g(t) - J(t) I < s}

(t EH, s > 0)

is also a topology since, for g E V1,.(f), with the notation 1J = s > 0, evidently

I g(t) -

J(t) I >

thus the sets Vr,.Cf) are open and their finite intersections furnish a neighbourhood base off consisting of open sets. We shall call it the topology of pointwise convergence. Further on we shall study in detail the topological spaces and topologies. 2.1.f. Exercises. 1. Let (xn) and (Yn) be two sequences, a and b the corresponding sequential filter bases. Prove that (a) if (xn) and (Yn) consist of the same members in a different order, then a "-' b; (b) if (yn) is a subsequence of (xn), then a < b; (c) the converse of the preceding statement is not true. 2. Prove that, if 2:f; (i E J) is an ascending system in E, then LJ and

n mi

mi

iE/

iEl

are of the same type. 3. Let and )8 be ascending systems in E. Show that n )8 = (u) )8, )8 c (n) )8 and in the latter formula equality does not hold in general. 4. Let &; (i EJ) be a filter in E. Prove that &i is a filter as well.

mu

m m

m

m

n

iEI

5. Let a and b be two filters in E. Show that a u b is not a centred system in general and it can occur that au b is centred but not a filter base. 6. Let A, c E (i EJ), A = LJ A;. Prove that A. = A.;.

n

iEI

iEl

7. Let A, B c E, C = Au B, D = An B. Prove that C =

A. (u) B, D =

= A. (n) B. 00

8. Show by an example that if & is a filter in E, Sn E & (n EN), then

n Sn = 0 1

can occur.

2.1.f.

EXERCISES

65

9. Let x = (xi, x 2) E R2 and, for s > 0, Vx,,= {(Yi,X2):[Yi-X1[ 0} is a neighbourhood base for a neighbourhood structure on E and that it is a topology if p satisfies (D 3). ll. For the metrics p' and p" defined in Exercise 1.3.h.4, show that

12. Let f be a function defined on [O, + oo ), increasing, continuous and concave (i.e. 0 < t < I implies J(tu + (I - t)v) ~ tf(u) + (1 - t)f(v)) for which f(O) = 0 and f(u) > 0 whenever u > 0. Show that, if p is a pseudo-metric on E and p'(x, y) = f(p(x, y)) for x, y EE, then p' is also a pseudo-metric and 8JP, = = 8Jp. [From the concavity off, it follows that f(v) - f(u) ~ f(w) - f(u) ~ f(w) - f(v) v-u w-u w-v

for O ~ u < v < w, i.e. f(u + v) ~ f(u) + f(v).] 13. We say that the sequence of real functions!,, defined on I "converges nearly everywhere" to the function f defined on I if there exists a countable set Mc I such that fn(t) ~ f(t) whenever t E I - M. Let Im; (m EN, i = l, ... , m) be the interval [ i

-:n l,

,

-! J,

gm;(t) = I for t E Im;, gm;(t) = 0 for t E I - Im;, and

let the sequence (In) consist of the functions 9mi in some order. Show that (a) (fn) does not converge nearly everywhere to any function; (b) from each subsequence of (fn) there can be selected a subsequence which converges nearly everywhere to f defined by the formula f(t) = 0 (t E I); (c) on the set E of real functions defined on l, there is no neighbourhood structure °'§! for which the convergence coincides with the convergence nearly everywhere. [If (hn) is a subsequence of(!,,) and hn{tn) = I, let Ctn) be a convergent subsequence of (tn).] 14. Let E be a finite set consisting of n elements. Show that (a) each filter in E is a principal filter; (b) there are 2n'-n different neighbourhood structures on E; (c) for n = 2, each neighbourhood structure on Eis a topology; (d) for n = 3, there is a neighbourhood structure on E which is not a topology.

5 Akos Csaszar

66

2.2.

DETERMINATION OF TOPOLOGIES

(2.2.1)

2.2. DETERMINATION OF TOPOLOGIES 2.2.a. Neighbourhood bases. A topology is, by definition, a neighbourhood structure where every point has a neighbourhood base consisting of open sets, briefly an open neighbourhood base. If we assign to each element x of a set E a filter b(x) as neighbourhood filter or a filter base b(x) as a neighbourhood base, the question arises, how to determine whether the resulting neighbourhood structure will be a topology or not. The following theorem gives an answer:

(2.2.1) Let a filter base b(x), consisting of subsets of E which contain x, be assigned to every element x of the set E, and let ~ be the neighbourhood structure on E which results from the filter bases b(x) as neighbourhood bases. ~ is a topology

if!

(V) for each set VE b(x) there is a V1 E b(x) such that every point y E Vi has a neighbourhood V2 E b(y) with the property V2 C V. Proof. If~ is a topology and VE b(x), then there is an open set GE b(x) such chat G c V (here b(x) denotes the neighbourhood filter of x). Let V1 E b(x) be such that Vi c G. If now y E Vi, then y is an interior point of G, hence V2 c G c c V for a suitable V2 E b(y). Conversely if (V) is fulfilled and V' E b(x), 1et V" be the set of all interior points of V'. If y E V" then V c V' for a suitable set VE b(y) and, applying then (V) to y instead of x, there is a V1 E o(y) such that V2 c V c V' for z E V1 and a suitable V2 E o(z). This shows that every point of Vi is an interior point of V', thus Vic V", and therefore y is an interior point of V". Since this holds for every pointy EV", the set V" is open and V" c V'. But V' E b(x) implies that x is also an interior point of V', thus x E V" and since V" is open, x is an interior point of V", hence V" E b(x). The open neighbourhoods of x constitute therefore a neighbourhood base of x. I If the neighbourhood filters of each point x are given, the condition (V) can be replaced by a simpler one:

(2.2.2) If a filter b(x) of E, consisting of sets which contain x, is assigned to each point x of E, let ~ be the neighbourhood structure for which iJ(x) is the neighbourhood filter of x. ~ is a topology if! (V') for each VE b(x), there is a Vi E b(x) such that VE b(y) whenever y E Vi. Proof. We get (V') from (V) because b(y) is an ascending system in E, on the other hand (V') implies (V) if b(x) = b(x). I It is advantageous for the determination of a topology to prescribe directly an open neighbourhood base of every point. The following theorem shows how to do this: (2.2.3) Let a filter base b(x), consisting of sets containing x, be assigned to each element x of E. These filter bases b(x) constitute open neighbourhood bases of the corresponding points for a topology 8T if! (V") for each VE b(x) and y E V, there is a Vi E b(y) such that V1 c V.

(2.2.6)

2.2.a.

NEIGHBOURHOOD BASES

67

Proof. If Vis open, then y E Vis an interior point of V and (V") is necessarily fulfilled. Conversely, in the neighbourhood structure gr determined by the filter bases b(x), V consists only of interior points in consequence of (V"), thus b(x) is an open neighbourhood base of x (and then gr is evidently a topology). I

Observe how the condition (V") is fulfilled in the case of the neighbourhood base {S(x,e) :e> O} of the topology grP of the pseudo-metric space [E, p] or of the neighbourhood base { [x, x

+ e)

: e > 0}

of the topology (6+. 2.2.b. Bases. It is extremely convenient to prescribe the neighbourhood bases of the points x EE in a set E in the following way: we give a system of sets Q3 consisting of subsets of E and prescribe that a neighbourhood base of x consists of all Sets containing X and belonging to Q}: (2.2.4)

b(x)

= {B: x EBE Q3}.

The following theorem corresponds to this: (2.2.5) Let Q3 be a system consisting of subsets of E. The system b(x) defined by (2.2.4) constitutes a neighbourhood base of x for a neighbourhood structure "'§! iff b(x) is a filter base for every x EE. "'§! is then a topology. Proof. By the definition of a neighbourhood base, o(x) has to be a filter base for every x. On the other hand, if this condition is fulfilled, (V") is also true (for Vi = V), thus "'§! is a topology according to (2.2.3). I

The following theorem answers the question, when does the above method furnish a given topology gr: (2.2.6) Let gr be a topology on E and Q) a system consisting of subsets of E. b(x) de.fined as in (2.2.4) will be a neighbourhood base of x for all x EE for the topology 8f iff (a) all sets BE Q3 are gr_open; (b) every non-empty gr _open set can be expressed as a union of sets belonging

tom. Proof. If (2.2.4) is a gr_neighbourhood base of x for every x, then every set B E Q3 is a neighbourhood of all of its points; thus it consists only of interior points and therefore it is open; moreover to each point x E G of a gr_open set G #- 0, there corresponds a Bx E Q3 such that x E Bx c G and evidently G

= U Bx. xEG

On the other hand, if (a) and (b) are fulfilled, then all sets BE o(x) of (2.2.4) are gr_neighbourhoods of x since they contain x and they are gr_open, and again 5*

68

2.2.

(2.2.7)

DETERMINATION OF TOPOLOGIES

every neighbourhood V of x contains an open neighbourhood G of x, and writing G as the union of sets belonging to )B, one of these members has to contain x: X

EB C

G C V,

B

E )B,

This shows that o(x) is a 8T-neighbourhood base of x.

I

A system )B with the properties (2.2.6) (a) and (b) is called a base for the topology 8T (or the topological space [E, 8T]). Using this term, we can formulate (2.2.6) and (2.2.5) in the following way: (2.2.7) Let 8T be a topology on E. A system of sets )B consisting of subsets of E is a base for 8T if.f, for x EE, the sets containing x and belonging to )B form a 8Tneighbourhood base of x. I (2.2.8) A system of sets )B consisting of subsets of E is a base for a suitable topology defined on E iJf, for each point x EE, the sets from )B containing x constitute a filter base. I E.g. we can see from 2.1.a (d) and (e) that all open intervals of the real line, or all open intervals with rational end-points form a base for the topology ©. It follows then from 2.1.a (i) and U) that all balls or all balls with rational radius form a base for the topology of any metric (and in the same way for any pseudometric) space. 2.1.a (m) shows that all open bricks form a base for @m, and (1.2.32) can be formulated by saying that the balls with rational radii and centres belonging to A form a base for ©m whenever Ac Rm is a dense set. Finally (1.3.11) can be formulated, straight away for pseudo-metric spaces, in the following way: (2.2.9) A pseudo-metric space is separable if.! it possesses a countable base.

I

Naturally the family of all open sets of a topology 8T fulfils conditions (2.2.6) (a) and (b). Accordingly: (2.2.10) All open sets constitute a base in any topological space [E, 8T], and all open sets containing x form a neighbourhood base for the point x EE. I The system of sets ® is said to be a subbase for the topology 8T if the finite intersections of sets belonging to 6 constitute a base for 8T; in other words ® is a system of 8T-open sets such that any non-empty 8T-open set can be expressed as the union of finite intersections of sets from ®· Hence we see that the topology can be given with the help of a sub base. The following theorem refers to this fact: (2.2.11) If ® is a covering of the set E, then ® can be considered as a subbase for a suitable topology on E. Proof. Denoting by Q3 the system consisting of all finite intersections of sets from ®, the sets from )B containing a given x EE constitute a filter base, so that )B is a base for a topology 8T on E according to (2.2.8). ® is a subbase for this topology 8T. II

(2.2.14)

2.2.c.

PRESCRIPTION OF OPEN OR CLOSED SETS

69

2.2.c. Prescription of open or closed sets. (2.2.10) shows that a topology is determined if we know the open or closed sets with respect to it (since the open sets are the complements of the closed ones). The question arises, what conditions has a system of sets ® or% to satisfy in order to be the system of open or closed sets, respectively, with respect to a suitable topology. The following two theorems give an answer: (2.2.12) A family of sets ® consisting of subsets of Eis identical with the family of all open sets of a suitable topology on E if! (G1) 0 E ®, EE ®; (G 2) G; E ® (i EI, I arbitrary) implies n

(G 3) G; E® (i = 1, ... , n) implies

LJ

G; E ®;

iEI

n G; E®· 1

Proof. We know from (2.1.30) that the family of open sets of any topology (moreover, of any neighbourhood structure) fulfils the conditions (G1)-(G3). On the other hand, if these are fulfilled by a system of sets @, then the sets belonging to ® and containing x constitute a filter base for any x EE, as x E EE ® and the intersection of two sets belonging to ® and containing x belongs to ® as well. Therefore, according to (2.2.8), ® is a base for a topology 3!. On account of (2.2.6) (a) and (b), the sets from® are 3!-open and, on the other hand, every Sf-open set is empty or the union of sets from ® thus belongs itself to ® in view of (G 2). II (2.2.13) The system % consisting of subsets of E is identical with the system of all closed sets for a suitable topology on E if!

(F1) 0E%, EE%; (F 2) F; E % (i EI, I arbitrary) implies n

n

F; E % ;

iEI

(F 3) F; E% (i = 1, ... , n) implies U F; E %1

Proof. (F1)-(F3) are fulfilled by the system % iff (GJ-(G 3) are true for the system %consisting of the complements of the sets from %- Hence by (2.1.30) (a) the statement follows. I E.g. let E = R and, for - CYJ ~ x ~ + (X), Gx = (- (X), x), (in particular G_oo = 0 for x = -(X), G+oo = E for x = +(X)). The system (2.2.14)

fulfils (G1)-(G3) since where

x = sup {X;: i E I} and

70 .

2.2.

DETERMINATION OF TOPOLOGIES

(2.2.15)

where

Therefore the system (2.2.14) is the system of open sets for a topology$. In the same way the system { (x,

+ 00): -

00 ~

X

~

+ 00}

is the system of open sets for a topology $ on R. As another example, let E '# 0 be an arbitrary set, and let ij consist of the set E and the finite subsets of E. Evidently (F1)-(F3) are fulfilled, thus ij is the system of closed ·sets for a topology. Denote this topology by gfE· 2.2.d. Interior and closure of a set. Let [E, ST] be a topological space, Ac E. The set of all interior points of A is called the interior of A and denoted by int A; the set of all limit points of A is called the closure of A and denoted by A. (2.2.15) Let [E, ST] be a topological space, A, B c E. Then (a) int A is the largest open set contained in A; (b) A is open if! A = int A; (c) Ac B implies int Ac int B; (d) int (An B) = int An int B. Proof. If x Eint A, there is an open neighbourhood V of x such that V c A, and since Vis a neighbourhood of y for y E V, it follows y Eint A, thus V c int A , so that all points x of int A are interior points of int A (not only of A). This shows that int A is open and naturally int A C A according to (2.1.28). On the other hand, if G c A is an open set then G is a neighbourhood of all points x E G, so that G c int A. (b) and (c) follow.immediately from (a). From (c), we get

int (A n B) c int A n int B and on the other hand, if x Eint A n int B, then A and B are neighbourhoods of x and the same is true for A n B so that x Eint (A n B). I (2.2.16) Let [E, ST] be a topological space, A, B c E. Then the following statements hold: (a) A = E - int(E - A); (b) A is the smallest closed set containing A; (c) A is closed if! A = A; (d) Ac B implies Ac B; (e) Au B = Au ii. Proof. E - A consists exactly of the exterior points of A, thus of the interior points of E - A according to (2.1.28). This proves (a). Hence we get (b) in view of (2.2.15) (a) since the smallest closed set which contains A is exactly the comple-

(2.2.17)

2.2.d.

INTERIOR AND CLOSURE OF A SET

71

ment of the largest open set contained in E - A. (c) and (d) follow immediately from (b). Finally we get (e) again from (a): Au B = E - int(E - (Au B)) = E - int((E - A) n (E - B)) =

=E

- (int(E - A) n int(E - B)) = (E - int(E - A)) u (E - int(E - B)) =

= l u B.

I

(2.2.15) (b) and (2.2.16) (c) show that the topology is determined if we know int A or A for each set A c E, since the equality A = int A determines the open sets and A = A the closed ones. We can again raise the question how the operation A --+ int A resp. A -+ A assigning subsets of E to the subsets of E is to be defined in order to coincide with the construction of interiors or of the closures. The following theorem gives us an answer with respect to the latter operation; the analogous theorem concerning int A could be easily formulated as well. (2.2.17) Let a subset A c E be assigned to each subset A c E. There exists a topology on E for which the closure of A is exactly A if! (K1) 0 = 0; (K2) Ac E implies Ac A;

A u B;

(Ks) A, B c E implies A u B = (K4) Ac E implies

= A =

A.

Proof. If A denotes the closure of A with respect to a topology, then, since 0 is closed and in view of (2.2.16) (c), (K1) is true, (K 2) holds by (2.2.16) (b), (Ks) is fulfilled according to (2.2.16) (e) and finally (K4) holds in view of (2.2.16) (b) and (c). Conversely, let us suppose that (K1)-(K4) are fulfilled and ~ is the system of those sets F c E for which F = P. Then (F1)-(Fs) are fulfilled. In fact, 0 E ~ according to (K1), and EE ~ according to (K 2). Whenever F1 = F; (i E J) and F = F;, then, by (Ks) and (K 2),

n

iEl

Pc

n P; = n F; = F c

iEI

P,

iEI

hence FE~ since (Ks) implies B =Au B =Au B:::, A for Ac B. If F1 = F1 n

(i = 1, ... , n) and D =

LJ F1,

then, according to (K3)

1

n

n

1

1

15 = LJ P, = LJ

F1

=

D.

Thus ~ is, in view of (2.2.13), the system of sets closed with respect to a topology /if. For arbitrary A c E, the set A is a closed set containing A in view of (K2) and (K4), namely the smallest one since, for A c F = P, A c P = F in view of the consequence of (K3) proved above. According to (2.2.16) (b), A is precisely the /if-closure of A. I

2.2.

72

(2.2.18)

DETERMINATION OF TOPOLOGIES

Let us now notice the following interesting property of int A and a consequence of (2.2.15) (a) and (2.2.16) (b):

A which is

(2.2.18) In any topological space, int A is the union of all open sets contained in the set A, and A is the intersection of all closed sets containing the set A. I Finally the following statement refers to pseudo-metric spaces: (2.2.19) If A =j:. 0 is bounded in the pseudo-metric space [E, p ], then A is bounded too and 8(A) = 8(A). Proof. For x, y EA and a given e > 0, let x 1 EA n S(x, e), y 1 EA n S(y, e). Then p(x, y) ;;;; p(x, x1)

+

p(x1 , Yi)

+ p(Yi, y) < e + 8(A) + e = 8(A) + 2e. 8(A) + 2e. From this 8(A) ;;;; 8(A) follows

Hence A is bounded and 8(A) ;;;; e --+ 0, and (1.2.14) implies 8(A) ;;;; 8(A).

I

for

2.2.e. The boundary of a set. We understand by the boundary of a set A, denoted (as an abbreviation of Latin "margo") by mar A, the set of all boundary points of the set A. According to (2.1.28):

(2.2.20) For any set A in the topological space [E, 8!], mar A = mar (E - A)= An E - A = A - int A;1 therefore mar A is always closed.

I

(2.2.21) For arbitrary sets A and B (a) mar (A u B) c mar A u mar B; (b) mar (An B) c mar Au mar B; (c) mar (A - B) c mar Au mar F. Proof. (a): According to (2.2.20), in view of int (A u B) ::> int A u int B which follows from (2.2.15) (c), by (2.2.16) (e)

mar (A u B) = A u B - int (A u B) c ( A u B) - (int A u int B) c c (A - int A) u (B - int B) = mar A u mar B. (b): Hence mar(A n B) = mar(E - (An B)) = mar ((E - A) u (E - B)) c c mar (E - A) u mar (E - B) = mar A u mar B. (c): Using now (b) we get mar (A - B) = mar (A n (E - B)) c mar A u mar(E - B) = = mar A u mar B. I We immediately get from (2.2.20), in view of (2.2.15) (b), (2.2.16) (c) and int Ac Ac A, the following statement:

(2.2.23)

2.2.e.

73

THE BOUNDARY OF A SET

(2.2.22) mar A = 0 if! A is a set simultaneously open and closed.

I

Such a set will be briefly called open-closed. 2.2.f. Axiomatic remarks. Tl:.e above results enable us to find different definitions, equivalent to ours, for the notion of topological space, and to construct the theory in a different way. E.g. we can say by definition that the set Eis a topological space if there is given a system @ of sets - called open sets - for which (GJ-(G 3) hold, or if there is given a system ij of sets · - called closed sets - for which (F1)-(F3) are true, or finally we can say that a set E is a topological space if an operation A - A is defined for the subsets of E, fulfilling (KJ-(K4). (2.2.12), (2.2.13), or (2.2.17) shows that any of these definitions is equivalent to our previous definition. According to these notions, conditions (G1)-(G3) or (F1)-(F3) or (KJ-(KJ can be considered as axioms in the theory of topological spaces (the latter are usually called Kuratowski closure axioms). The way we introduced the notion of a topological space had the advantage that it started from the notion of neighbourhood or neighbourhood base according to the fact that the topologies are mostly defined in this way (e.g. in the case of metric spaces) or sometimes by prescribing a (sub)base and it seldom happens that the open or closed sets or the closures of sets are primarily given. On the other hand, it is true that the latter variants of the definition of a topological space have a simpler form; therefore we obtain a more rapid development of the theory by using them. If we define e.g. a topological space with the help of the Kuratowski axioms, then first the closed sets will be defined by the equality A = A, then the open sets as the complements of the closed sets, and finally the neighbourhoods of the point x as sets containing an open subset which contains the point x. It is still worth formulating the definition of a topological space which is based on prescribing open neighbourhood bases:

(2.2.23) A topological space is a set E in which, to each x E E, there is given a system b(x) of subsets of E with the properties: (H1) b(x) ::/= 0; (H 2) xEVfor VEb(x); (H 3) if Vi, V2 E b(x), then there is a Va E b(x) such that V3 C Vin V2 ; (H4) if VE b(x) and y E V, there is Vi E b(y) such that Vi C V. In fact, (HJ-(H 3) describe the fact that b(x) is a filter base consisting of sets which contain the point x, and (H4) is equivalent to the condition (V") under (2.2.3). I The conditions (HJ-(H4) are called Hausdorff neighbourhood axioms. 2.2.g. Exercises, 1. Let E be the set of integers. Denote by Mx the set of the multiples of x EE, and l.1(x) = M x• Prove that the neighbourhood structure thus obtained ,is a topology.

74

2.2.

DETERMINATION OF TOPOLOGIES

(2.3.1)

2. Let E = Rand let SJ consist of all intervals having the form (- oo, x) and (- oo, x] (x ER) and moreover of 0 and R. Show that SJ can be considered a!> the system of all open sets of a topology and, at the same time, of all closed sets of another topology. ,3. Let Q3 be a system of subsets of E such that Bi, B 2 EQ3 implies B1 n B 2 E Q3 and E = U {B: B EQ3}. Prove that Q3 can be considered as the base of a topology on E. ·4. Let Q3 consist, in the plane R2, of the straight lines going through the point (O, 0) = a and cif the set {a}. Show that Q3 is a base for a topology on R2• 5. Let [E, 81"] be a topological space and denote by ~(A), for A c E, the system of non-empty subsets of the set A. Show that the sets ~(G), where G runs over the 81"-open sets, constitute a base for a topology on ~(E). 6. Show that the intervals [a, b) (a < b) constitute a base for @+, and the intervals (a, b] constitute a base for@-'-. 7. Show that we obtain a topology on any set E if we consider as closed sets the countable subsets of E together with E. 8. Let E ,be a set of more than one element, ordered by the order relation < and, for x EE, (a) (+-, x) = {y: y < x}; (b) (+-, x] = {y: y ~ x}; (c) (x, --+) = {y: x < y}; (d) [x, --+) = {y: X ~ y}. Show that the sets of the forms (a) and (c) or the sets of the forms (a) and (d) or the sets of the forms (b) and (c) constitute a subbase for a topology on E. We call the first of them the order topology of E. Into what do these topologies go over if E = R with the usual relation < ? Prove that, besides the sets figuring in the subbase, the sets of the form (a, b) = {x: a < x < b }, or [a, b) = {x: a ~ x < b }, or (a, b] = {x: a< x ~ b} respectively constitute a base for the above topologies. 9. Let 8i be the order topology of the ordered set E. By defining the terms of lower and upper bound, bounded· from below or above with respect to the set A c E in the same way as in the case of E = R, prove that the least upper bound and the greatest lower bound of A ,f= 0 belong to the 81"-closure of A. 10. Show that, in the space [Rm, @m] (a) intS(a, e) = S(a, e); (b) S(a,e) = S(a,e); (c) int [a1, ... , am; b1,.,,, bm] = (ai, . . , am; bi,.,,, bm); (d) (ai, ... , am; bi, ... , bm) = [a1, . . . , am; bi, ... , bm]; if a ERm, e > 0, a1 < b1 (i = 1, ... , m).

2.3.a.

(2.3.2)

75

COMPARISON OF TOPOLOGIES

I 1. Look for a metric space [E, p] in which the equalities 10 (a) and (b) are not always true. 12. Let [E, p] be a pseudo-metric space, A, B c E, A =/. 0 =/. B. Show that p(A, B) = p(A, B). 13. Let [E, ST] be a topological space and let us consider the following statements for A,BcE: (a) A is the interior of a closed set; (b) A= intA; (c) B is the closure of an open set; (d) B = intB. Show that (a) and (b), as well as (c) and (d) are equivalent to each other, and (a) is valid for A iff (c) holds for E - A = B. Give an example in the space [R, $] of an open (closed) set for which (a) ((c)) holds, and for sets for which these statements are not true. 14. In a topological space [E, 8f] let A be the system of sets whose boundary is finite (countable). Prove that A, BE 2timplies Au BE 2'C, An BE 2'C, A - BE 2r. 2.3. COMPARISON AND RESTRICTION OF TOPOLOGIES 2.3.a. Comparison of topologies. Let "\91 and "\92 be neighbourhood structures on the set E and suppose that the neighbourhood filter tJ 1(x) with respect to "\91 of any point x EE is coarser than the neighbourhood filter tJlx) with respect to "\92of the same point: tJ1(x) < tJlx). Then we say that the neighbourhood structure "\91 is coarser than "\92 or that "\92 is finer than "\91; we denote it by

"'91 < "'§'2 or "'92 > "\91. Instead of tJ 1(x) and tJlx), we can evidently consider filter bases equivalent to them, i.e. neighbourhood bases o1(x) and blx) of x with respect to "\91 and "\92 respectively. Accordingly "\91 < "'§'2 iff any neighbourhood V1 E o1(x) from the neighbourhood base 1(x) of any point x EE has a subset V2 E 2(x). E.g . .$ b(x)

that is to say if every neighbourhood of x contains a subset from t. Instead of b(x), any other filter base equivalent to it, i.e. an arbitrary neighbourhood base of x, can evidently be considered here. Consequently the convergence of filter bases defined in this way is a generalization of the convergence of sequences of points. We can formulate now the following statement corresponding to (1.2.26): (2.4.2) Let [E, °\9] be a neighbourhood space, A c E, x EE. x is a limit point of the set A. if! there is a filter base in A converging to x. A is closed if! x EA follows from the fact that t is a filter base in A and t -+ x. Proof. If t is a filter base in A and t -+ x, then every neighbourhood of x contains a set from t, hence a non-empty subset of A, and therefore intersects A. On the other hand, if xis a limit point of A, then, by 0 ~ b(x)(n) {A}, t = b(x) (n) {A} is a filter base in A and t -+ x, since b(x) < t by virtue of (2.1.21). The second part of the statement follows immediately from the first part by (2.1.29).1 We obtain at once from the definition: 6*

2.4.

84

(2.4.3)

CONVERGENCE OF FILTER BASES

(2.4.3) lf in a neighbourhood space [E, "\9], r---+ x and t 1 is a filter base in E finer than r, then t 1 ---+ x. Equivalent filter bases are,from the point of view of convergence, equivalent to each other. I This is a generalization of (1.2.5) since it is evident that the sequential filter base belonging to some subsequence of the sequence (xn) is finer than the sequential filter base belonging to (xn). In the same way, the following theorem is a generalization of (1.2.4): (2.4.4) The fundamental filter base x belonging to any point x EE of the neighbourhood space [E, "\9] converges to x. I On the other hand, the next theorem corresponds to (1.2.6): (2.4.5) If, in the neighbourhood space [E, "\9], t 1

x, t 2 ---+ x, then t 1 (u) t 2 ---+ x.

---+

Proof. We know from (2.1.23) that, together with r 1 and r 2 , r 1 (u) t 2 is a filter base in E. Each neighbourhood of x contains a subset R 1 E r1 and a subset R 2 E r2 and consequently also the set R 1 u R 2 E r 1 (u) r 2 . I (2.4.6) Let "\91 and "\92 be neighbourhood structures on E, "\91 < "\92 . If t ---+ x with respect to "\92 then r ---+ x with respect to "\91 as well. Proof. Denoting the neighbourhood filters as usual, we have r > blx) > tJi(x) by hypothesis. I If it is necessary to indicate that the filter base

r converges to

x with respect to

a neighbourhood structure "\9, we can use the notation

(2.4. 7) If 8!; (i EJ) is a topology on E and r ---+ x (8!;) for each i E J, then r ---+ x (8!) with the notation 8f = sup {8!;: i El}.

Proof. According to (2.3.7), the sets of the form

where x E G;k and G;k is 8l;k-open for ik E J (k = I, ... , n) constitute a 8!-neighbourhood base of x. By hypothesis, there is, for every k, a set Rk E r such that n

Rk

c

G;k

(k = I, ... , n), then RE r such that R

c

n Rk·

Hence R

c

V.

I

1

(2.4.8) Let [E, "\9] be a neighbourhood space, 0 # E 0 c E, r a filter base in E0 , x EE0 • Then

r ---+ x

hold simultaneously.

("\9) and r ---+ x

("\9 I Eo)

(2.4.12)

2.4.c.

85

AXIOMS OF COUNTABILITY

Proof. As r is a filter base in E 0 , r

< {E0 } is valid. Thus r >

b(x) implies

r (n) r > b(x) (n) {E0 }

and since r is a filter base, by (2.1.19) r > r (n) r. On the other hand, r > > b(x) (n) {E0 } clearly implies r > b(x), since b(x) (n) {E0 } > b(x), I 2.4.c. Axioms of countability. It may be asked why the closed sets of a metric space can be characterized by the convergence of sequences of points, while this is not possible in a topological space in general. The kernel of the question is the following simple remark: (2.4.9) If a point x EE has a countable neighbourhood base l.J(x) in a neighbourhood space [E, °\9] and xis a limit point of the set ACE, then there is a sequence (xn) in E converging to x. n

Proof. Let {Vn: n = I, 2, ... } = l.J(x), V~ =

n V; is a neighbourhood of x as i

well, thus V~ n A =I= 0; suppose xn E V~ n A. Then the sequence (xn) is the required one. In fact, if V is an arbitrary neighbourhood of x, then Vn, c V for some n0 and of course n ~ n0 implies xn E V~ c Vn, c V. I Let us call the following condition the first axiom of countability: (Mi)Each point has a countable neighbourhood base in the topological space [E, 3r].

We shall call the topology 8f and the topological space [E, 8f] satisfying (Mi) an M 1-topology and an Mi-space respectively. · In an Mi-space, the following sharpening of (2.4.2) is true: (2.4.10) Let [E, 3r] be an Mi-space, A c E, x c E. x is a limit point of the set A (ff there is a sequence of points in A which converges to x. A is closed iJJ xn EA, Xn -+ x implies x EA. Proof. If x is a limit point of A, then by (2.4.9) there is a sequence of points in A which converges to x; conversely if there exists a sequence of points of this kind, then applying (2.4.2) to the sequential filter base belonging to it, we find that x is a limit point of A. From this, we obtain the second part in view of (2.1.29). I

In metric and, more generally, in pseudo-metric spaces, theorem (2.4.10) can be applied since in such spaces the balls

s(x, : )(n EN)

constitute a neigh-

bourhood base for x: (2.4.11) Every pseudo-metric space is an Mi-space. I It is often useful to know that a countable neighbourhood base of a point can be subjected to further restrictions. The following two theorems deal with this fact:

(2.4.12) In an Mi-space, a countable neighbourhood base can be selected from any given neighbourhood base of a point.

2.4.

86

(2.4.13)

CONVERGENCE OF FILTER BASES

Proof. Let {Vn: n EN} be a countable neighbourhood base of x and o(x) an arbitrary neighbourhood base. Choose V~ E o(x) such that V~ c Vn. It is evident that {V~: n EN) is a neighbourhood base of x. I

Accordingly, in an Mi-space any point has a countable open neighbourhood base. Moreover: (2.4.13) In an Mi-space, any point has an open neighbourhood base {Vn: n EN} such that V 1 :::) V2 :::) V3 :::) . • • • Proof. In view of (2.4.12), x has an open countable neighbourhood base

{v~: n E N}.

n

Let Vn =

n v;. The system { vn: n EN} fulfils the requirements. I 1

The following second axiom of countability contains stronger restrictions than the axiom (M 1): (M 2) In the topological space [E, 81"], there is a countable base.

We call a topology 8T or a topological space [E, 81"] satisfying (M 2) an M 2topology or an M 2-space respectively. It follows at once from (2.2.6) that: (2.4.14) Every M 2-space is an M 1 -space.

I

Theorem (2.2.9) can evidently be formulated as follows: a pseudo-metric space is separable iff it is an M 2-space. This can be expressed somewhat more generally. Let us call dense (or everywhere dense), as in the case of metric spaces, a subset A c E of the topological space [E, 81"] if A = E, that is to say, ifany neighbourhood of any point of E intersects A. We say that the space [E, 81"] is separable if there is a countable dense set in it. These definitions are naturally generalizations of the corresponding definitions formulated for metric spaces. Now one part of (2.2.9) is true in general : (2.4.15) Every M 2-space is separable. Proof. Let {En: n E N} be a base and Xn E En (we can suppose that the sets En are not empty). A = {xn: n E N} is then dense, for any neighbourhood V of any point x contains by (2.2.6) a set En and with it a point Xn EA. I

On the other hand, in view of (2.2.9): (2.4.16) A separable pseudo-metric space is an Mrspace.

I

The following theorem corresponding to theorem (2.4.12) is also true: (2.4.17) In an M 2-space, a countable base can be selected from any base )8. Proof. Let {Bn: n EN} be a countable base and consider those pairs of indices (i,j) for which B; c B c Bj for a suitable set BE 58. There are countably many such pairs (i,j). Choose for each pair of this kind a set Bu E )8 satisfying the condition B c Bu c Bi.

(2.4.21)

2.4.d.

EXAMPLES. METRIZABLE SPACES

87

The sets selected in this way constitute the required base which is evidently countable. In fact, if G is open and x E G, then there is a Bi for which x EBi c G, then a B E 58 such that x EB c "B.i, and finally a Bi such that x EBi c B. The pair (i,j) obtained in this way is one of the selected ones, thus x EB 1 c Bii c Bi cG, I Lindelof's theorem (1.2.34) can be extended to M 2-spaces. For this purpose, repeating word for word the definition of an (open, closed, finite, countable) covering for an arbitrary topological space, we must note that we made use, in the proof of (1.2.34), only of the fact that (Rm, $m) is an M 2-space. Let us agree to call a topological space [E, 81"] a Lindelof space (and the topology 81" Lindelof topology respectively) if from each open covering of E there can be selected a countable covering. Then we can state: (2.4.18) Every M 2-space is a Lindelof space.

I

Let us notice finally as an obvious consequence of (2.3.10) and (2.3.13) (e): (2.4.19) A subspace of an Mi-space is an Mi-space; a subspace of an M 2-space is an M 2-space as well. I

In the same way, we get from (2.3.6) and (2.3.7): (2.4.20) If I is countable and 8ri is an Mi- (M2-)topology for i EI, then the same is true for sup {8ri: i EI} too, I

2.4.d. Examples. Metrizable spaces. In section 2.4.a, we have seen an example of a topological space in which the statement of theorem (2.4.10) does not hold. Thus this space is not an Mi-space. Every non-separable metric space is an example showing that axiom (M 2) is strictly stronger than axiom (MJ (see (2.4.11) and (2.4.15)). A further important example is the following one: (2.4.21) The space [R, $+] has the following properties; (a) it is an Mi-space; (b) it is not an M 2-space; (c) all its subsets are separable; (d) all its subsets are Lindelof spaces.

Proof. (a): The intervals [ x, x

+

!)

(n EN) evidently constitute an $+-

neighbourhood base of x. (b): As e.g. [x, x + 1) is an $+-neighbourhood of x, it is clear that in each base of the space there has to be a set whose smallest element is precisely x. This cannot be true for a countable base as the set R is not countable. In order to prove (c) and (d) we shall need the following statement: (*) Any set A c R contains at most countably many points x EA such that there is an n

EN with the property (x, x +

!)

nA

= 0.

2.4.

88

(2.4.22)

CONVERGENCE OF FILTER BASES

In fact, denoting by Bn the set of those x EA for which (x, x

+ ~ )nA

= 0

holds, any interval [: , k : 1 ) (k is an integer) contains at most one point of

Bn- Thus Bn is countable. (c): Suppose 0 =I- Ac E. By (2.4.19) &, I A is an M 2-topology; thus there is, according to (2.4.15), a countable (&, I A)-dense set Mc A. If x EA does not belong to the exceptional countable set under(*) then, for any &,+-neighbourhood V of x, there is an n EN for which [ x, x

n (x, x

+ ~ ), finally

a z EM n ( x, x

+

+ ~)c

V, then a point with y E A n

!)

c Mn V. Therefore if we add to M

the countably many exceptional points mentioned above, the set M' c A obtained will be (&,+ I A)-dense. (d): Let 0 =I- E0 , {G; n E 0 : i EI} be a covering of E0 and G; &,+-open for every i. For each x EE 0 , there is then an i EJ for which x E G;, then an Bx > 0 such that [x, x + ex} c G;. Let A :J E 0 be the union of the intervals [x, x + ex} obtained in this way and B = A -

LJ

(x, x

+ ex) .

xEE,

Every pointy EB lies in an interval [x, x

+

Bx)

and choosing then a number n E N such that y +

+

2-)n

nB

= 0. Thus

+

(it coincides with its left end-point)

< x + Bx, evidently

B is countable. On the other hand the set

U

(Y, y +

(x, x

+

ex)

x~

can be covered, according to the Lindelof theorem (1.2.34), with countably many intervals [xk, xk + Bxk) (k EN); selecting for each interval [xk> xk + Bxk) a set G;k (ik E J) containing it and then choosing for each point of B n E0 a set G, (i E J) containing it, finally the sets G;k together with the countably many sets G; now chosen cover the set A; thus their intersection with E0 covers the setE0 by A:JE0 • I It was not superfluous to emphasize in the above example that all subspaces of the space are separable, for the subspaces of a separable space need not be separable. The following example will show this: (2.4.22) If E Q(x, e)

= R 2 and 8! is the topology for which the sets = {(Y1, Y2):

X1 ;::;;

Yr


0. (c) Choose for every point x ~ E0 a number Bx> 0 such that Q(x, Bx) n E0 = 0, and for the points x EE0 an arbitrary Bx > 0. The sets Q(x, Bx) obtained in this way give an open covering of E such that no countable subsystem can cover E0 • I

We now come to the following statement: (2.4.23) Let the set H =f. 0 be arbitrary, let Ebe the set of real functions de.fined on H, gr the topology of pointwise convergence. If H is not countable, then [E, gr] is not an Mi-space. Proof. In view of (2.4.12) it is enough to show that, with the usual notation

v,,.

= {g; I g(t) - f(t) I < B} (t E H,

B

> 0) ,

no countable neighbourhood base can be selected from the neighbourhood base off EE consisting of the sets of the form

(2.4.24) However, in each set (2.4.24), a finite number of ti EH occurs; thus we can find for a countable system of them a t EH which occurs in none of them. Therefore, for some B > 0, Vi,. certainly does not contain any set of the subsystem in question, I From our examples we see that some topological spaces display phenomena which cannot occur in metric or pseudo-metric spaces: the space under (2.4.23) is not an M1-space although each pseudo-metric space has this property according to (2.4.11). The topology 16+ is separable without being an M 2 -topology in contradiction to (2.4.16). The space (2.4.22) is separable without being a Lindelof space, in contradiction to (2.4.16) and (2.4.18). Thus if we agree to say that a topology gr or a topological space [E, grJ is metrizable (pseudo-metrizable) if a metric (pseudo-metric) p can be introduced on E such that gr is identical with grp, then we can say that none of the spaces mentioned above is pseudo-metrizable. Hence it is clear that non-metrizable topologies occur in connection with relatively elementary questions of analysis (convergence of numerical sequences from one side, pointwise convergence of sequences of functions). 2.4.e. Exercises. 1. Prove that the system of sets ~ in the set E is an ideal if (a) Eis arbitrary, ~ is the system of finite subsets; (b) E is a metric space, ~ is the system of bounded sets; (c) Eis a topological space, ~ consists of all sets which can be covered with the boundary of an open set; (d) Eis a topological space, ~ consists of the sets which can be covered with the boundaries of countably many open sets.

90

2.4,

(2.5.1)

CONVERGENCE OF FILTER BASES

2. Show that if ~ 1 and ~ 2 are ideals consisting of subsets of the topological space [E, 8T] and ~ 1 c ~ 2 then § 31 < 8T3 , with the notation of (2.4.1), but the latter relation does not imply ~ 1 c ~ 2 in general. [E = R, 8f = $, ~ 1 consists of the finite sets, ~ 2 of the empty set only.] 3. If [E, '\9] is a neighbourhood space, t a filter base in E, x EE, suppose that t does not converge to x. Show that there exists then a filter base t 1 finer than t such that no filter base finer than t 1 converges to x. 4. Give an example for a set E and topologies 8f1 and W2 on E where § 1 is an Mrtopology but W2 has not this property, such that W1 < W2 , and also such that 8'f2 < Wi. [Every indiscrete and every discrete space is an Mrspace.] 5. Show that the space [E, 8FE] (a) is an M 2-space if E is countable; (b) is not an Mi-space but is separable if Eis uncountable. 6. Show that the two topologies defined in Exercise 2.2.g.2 are separable Mitopologies but are not M 2-topologies. 7. Give an example of an Mi-space which is not separable. 8. Prove that if Wiand 8'f2 are topologies on E, W2 is separable, 8'f1 < Sf2 , then W1 is separable as well. · · 9. Let ~ consist of those sets on R which can be covered with the boundary of an $-open set and, with the notation (2.4.1), let 8f = $3 . Show that (a) if x E Q, then no (8f I Q)-neighbourhood of x consists of x only; (b) if Vn (n EN) is a (8f I Q)-neighbourhood of the point x E Q and Xn E Vn satisfies O < I Xn - x

I < _!_ n

then R - {xn: n EN} is 8f-open;

(c) 8f I Q is not an M 1 -topology. 10. Give an example for topologies 8ii and 8f2 on a set E such that 8ii < 8f2, &f2 is an M 2-topology but 8f1 is not an Mi-topology. · 11. Give an example of separable topologies 8f1 and 8f2 on a set E such that sup {8'f1, 8f2 } is not separable. [$+ and $- .] 12. Prove that if each one of the topologies 8f1 < 8f2 < 8f3 < ... is separable, then 8f = sup {8fi: i EN} is separable as well. 13. Let ~ be the system of the countable subsets of Rand, with the notation (2.4.1), let 8f = $3 . Show that 8f is (not M 1- and) a non-separable topology but is a Lindelof topology. 00

[If x ER, x E Gx - Mx, where Gx is @-open, Mx Eoil then R =

00

U Gx1 and U Mx 1 1

1

is countable.] 14. Prove that if 8ii < 8f2 are two topologies and 8f2 is Lindelof, then 8f1 is a Lindelof topology as well. 15. Let E = R, let 58 consist of the sets E - {1, 2, ... , n} and the sets {x} where x =/: 0. Show that

2.5.a,

(2.5.1)

(a) (b) (c) (d) (e)

Q3 is a base for a topology

gr gr gr gr

91

BASIC NOTIONS

gr;

is a Lindelof topology; is an Mi-topology; is not separable; E - {O} is not Lindelof.

J

16. Give an example for Lindelof topologies gr1 and gr2 on a set E such that sup {gri, gr2 } is not Lindelof. 17. Let E = R, and let Bn (n EN) consist of the subsets having finite complements, and of the sets {x} where x =I= n, n + 1, n + 2, .... Show that (a) (b) (c) (d)

Q3n is a base for a topology &fn;

grl < &f2 < • • .; grn is Lindelof; gr = sup {&rn: n EN} is not Lindelof.

18. Prove that if a pseudo-metric space is Lindelof, then it is an M 2-space. [Choose a countable covering {S (Xni,

N} from the covering {Slx, : ) :

!J:

iE

N}.J

x EE} arid consider the set {xn1: n EN, i E

2.5. SEPARATION AXIOMS 2.5.a. Basic notions. We have already mentioned in 2.1.e that in a neighbourhood space a sequence may converge to more than one limit point in contrast to metric spaces where this is impossible in view of (1.2.8). The same is true for filter bases. Accordingly, we can easily prove the following remark:

(2.5.1) Let [E, '19] be a neighbourhood space and x, y t in E which converges simultaneously to x and to y

EE. We can find a filter base if! each neighbourhood of x

intersects each neighbourhood of y. Proof. If t ---t x and t -> y, then t > tJ(x), t > tJ(y), thus by (2.1.19) and (2.1.16) t > t (n) t > tJ(x) (n) tJ(y). Hence every set of tJ(x) (n) tJ(y) contains then a subset belonging to t and is therefore non-empty. On the other hand, if 0 ~ tJ(x) · (n) tJ(y), then by (2.1.21) t -> x and t -> y is fulfilled by the filter base t = = tJ(x) (n) tJ(y) itself. I

Accordingly, the presence of filter bases converging to several limit points can be excluded by requiring that two different points always have disjoint neighbourhoods. Similar conditions occur in various forms; they are commonly called separation axioms. , In order to formulate these conditions more easily it is useful to introduce the following terms in the case of topological spaces.

92

2.5.

SEPARATION AXIOMS

(2.5.2)

We understand by a neighbourhood of the set Ac E, in a topological space·· [E, 3f], a set V for which A c int V. Evidently the neighbourhoods of the set {x} consisting of a single element are identical with the neighbourhoods of the point x. (2.5.2) In a topological space [E. 3i], the neighbourhoods of any set 0 =fa A c E constitute a filter in E, called the neighbourhood filter of A and denoted by l:J(A). We call the bases of the filter l:J(A) neighbourhood bases of A. Each set 0 =fa Ac E has an open neighbourhood base that is a neighbourhood base consisting of open sets only. Proof. It follows from (2.5.15) that l:J(A) is a filter in E and that the system of sets {int V: VE l:J(A)} is a base of l:J(A) consisting of open sets. I

In the case of A = {x} the notion of (open) neighbourhood base coincides with the notion of (open) neighbourhood base of the point x. Now the non-empty sets A and Bare said to be disconnected in a topological space if they have disjoint neighbourhoods, i.e. if 0 E l:J(A) (n) l:J(B); then by (2.5.2) A and B have disjoint open neighbourhoods as well. A and B are said to be separated if A has a neighbourhood disjoint from Band B has a neighbourhood disjoint from A; A and B are weakly separated if at least one of them possesses a neighbourhood not intersecting the other one. Evidently the neighbourhoods figuring here can always be chosen from an arbitrary neighbourhood base of A respectively B, e.g. they can always be chosen to be open. Hence we get the next remark: (2.5.3) In the topological space [E, 3i] A possesses a neighbourhood not intersecting B if! A n jj = 0. Thus A and B are separated if! A n jj = A n B = 0 (A, B =fa 0). Proof. If A c int V and V n B = 0, then B c E - int V since the set int Vis open, E - int V is closed, thus B c E - int V and A n jj = 0. On the other hand, if A n jj = 0, then the open set E - B is a neighbourhood of A not intersecting B. I In view of (2.5.3) the sets A =fa f,J, B =fa 0 are said to be strongly separated if B are either disconnected or strongly separated, then they are separated, and if they are separated, then they are weakly separated as well. Applying the above notions to sets {x} of a single element, we shall speak of the point x; e.g. we say that the point x and the set A =fa 0 are disconnected if 0 E l:J(x) (n) l:J(A). We can extend the former definitions to the case where one of the sets A and B (or both) are empty; as 0 itself is a neighbourhood of 0, 0 and any set A c E are disconnected and strongly separated.

A n B = 0. Evidently, if A and

2.5.b, T0-spaces, F'irst let us consider the following separation axiom:

(T0) In the topological space [E, 3i] any two points x =fa y are weakly separated.

2.5.b.

(2.5.9)

T 0 -SPACES

93

We shall call T0-space or T0-topology a topological space or a topology satisfying this axiom and we shall use similar terminology in connection with other separation axioms. The following remark enables us to formulate axiom (T0) in another manner: (2.5.4) In a topological space two points x and y are weakly separated iff v(x) # # v(y) for their neighbourhood filters. Proof. If e.g. VE o(x) and y ~ V, then evidently V ~ v(y). On the other hand, if tJ(x) # v(y), e.g. VE o(x), V ~ b(y), then int VE o(x) and y ~ int V since the open set int V is a neighbourhood of any of its points. I Any pseudo-metric space which is not a metric space gives an example of a topological space which is not a T0 -space. In fact §

(2.5.5) A pseudo-metric p given on the set E is a metric on E iff the topology is a T0-topology. Proof. p(x, y) = 0 holds iff y E S(x, e) and x E S(y, e) for every e > 0. I

P

(2.5.6) Any subspace of a T0-space is a T0-space. Proof. If 0 # E0 c E, x, y EE0 , x # y, then there is e.g. VE o(x) such that y~ V. Then V0 = VnE0 E v(x) (n) {E0 } and y~ Vo, I (2.5.7) If gr is a T0-topology on E and gr < gr, is another topology, then gr' is a T0-topology as well. Proof. If x, y EE, x # y, then e.g. x has a gr_open neighbourhood not containing y. This is §' -open as well. I 2.5.c. Ti-spaces. Let us consider the following axiom:

(SJ In the topological space [E, §] any two weakly separated points are separated. We can reformulate this with the help of the following remark: (2.5.8) The following statements are equivalent for a topological space [E, §]: (a) [E, gr] is an Si-space; (b) any neighbourhood of any point x contains the closure x of x (i.e. the closure {x} of the set {x}); (c) any point x and any closed set F not containing x are separated. Proof. (a) => (b): If VE o(x) and y EE - V, then x and y are weakly separated, hence separated, and by (2.5.3) y ~ x. Thus x c V. (b) => (c): If x ~ F = F, thenE - FE v(x) implies x c E - F, so that x n F = = x n P =;= 0. Therefore x and Fare (strongly) separated. (c) (a): If x and y are weakly separated, then e.g. x has an open neighbourhood G not containing y. Then x and E - Gare separated; thus x n (E - G) = 0 and, by y E E - G, x n {y} = 0, finally ji c E - G implies {x} n ji = 0. I

=

(2.5.9) Any subspace of an Si-space is an Si-space.

94

2.5.

(2.5.10)

SEPARATION AXIOMS

Proof. Let [E, W] be an Si-space, 0 =I= E 0 c E, x, y EE 0 and suppose that x and y are weakly separated with respect to W [ E 0 • Then by (2.5.3) e.g. (x n E0 ) n n {y} = 0; thus x n {y} = 0 and from this {x} n y = 0; finally a fortiori {x} n (y n E0 ) = 0. I We get the following axiom as the union of axioms (T0 ) and (S1): (T 1) In the topological space [E, 8i] any two points x =I= y are separated.

Accordingly: (2.5.10) A topological space is a Ti-space

if! it

is a T0-space and an Si-space.

I

Another characteristic property is the following one: (2.5.11) A topological space is a T 1-space if! each subset consisting of one point is closed. Proof. In a Ti-space x =I= y implies y ~ x by (2.5.3), thus x = {x}. On the other hand, if every set of one element is closed, then, for any two points x =I= y, x n y = 0 holds, and the given points are (strongly) separated. I A consequence of (2.5.6), (2.5.9) and (2.5.10) is the following: (2.5.12) Any subspace of a Ti-space is a Ti-space.

I

The following statement corresponding to (2.5. 7) is also true: (2.5.13) If W is a Ti-topology on E and 8i < W' is another topology, then W' is a Ti-topology as well. Proof. If every set of one element is W-closed, then it is W'-closed as well and we can apply (2.5.11). I (2.5.14) If W; is an S 1-topology on E and W = sup {W;: i EI}, then 8i is an S 1 topology as well. Proof. Suppose that G is a W-open set which contains x but does not contain n

y. Then x E

n G;; c

G where G;; is W;topen and

~

E I. There is a j with y

~

1

and hence a W;topen and a fortiori W-open set H;; such that x

~

H;;, y E H;r

G;

I

It is easy to see that the topology ~ satisfies (T0) but does not satisfy (T1). Thus it is not an Si-topology either. The following theorem gives an example for an Si-space which is not a T0-space : (2.5.15) Every pseudo-metric space is an S 1-space. . Proof. If x and y are weakly separated, then for some e > 0 e.g. y ~ S(x, e), from which x ~ S(y, e). I

2.5.d. T2-spaces. As two disconnected sets are separated, a stronger condition than (S 1) is the following one :

(2.5.23)

2.5.d,

95

T2-SPACES

(S 2) In the topological space [E, gr] any two weakly separated points are disconnected. According to the above remark: (2,5.16) Any S 2-space is an S 1-space.

I

We get the following axiom by joining (S 2) and (T0): (T2) In the topological space [E, gr] any two points x #= y are disconnected. The T2-spaces are also called Hausdorff spaces (or separated spaces). According to the foregoing: (2.5.17) A topological space is a T 2-space iff it is a T0 -space and an S 2-space.

I

From (2.5:10), (2.5.17), and (2.5.16), we get: (2.5.18) Every T 2-space is a Ti-space,

I

We obtain from (2.5.1) the following important characterization of T2-spaces: (2.5.19) A topological space is a T 2-space iff any filter base in it has at most one limit point. I

It holds similarly to (2.5.9): (2.5.20) Any subspace of an S 2-space is an S 2-space. Proof. Let [E, 8T] be an S2-space, 0 #= E0 c E, x and y weakly separated with respect to gr I E0 • Then e.g. (x n E0) n {y} = 0, thus x n {y} = 0 and there are V1 E b(x), V2 E b(y) such that Vin V2 = 0. Vin E0 and V2 n E0 are then disjoint ($ I E0)-neighbourhoods of x and y respectively. I We get from (2.5.17), (2.5.20), and (2.5.6): (2.5.21) Any subspace of a T 2-space is a T 2-space.

I

Similarly to (2.5.13), it is true that (2.5.22) If gr is a T 2-topology on E and gr < gr, is another topology, then gr, is a T2 -topology. Proof. If x #= y, then x and y have disjoint gr_open neighbourhoods. These neighbourhoods are $' -open as well. I Similarly to (2.5.14), we can prove: (2.5.23) If 8T; is an S 2-topology on E (i E[) and gr= sup {81";: i EI} then 8T is an S 2-topology as well. I The topology 8FE, in the case of an infinite fundamental set E, gives an example of a T1-topology which is not a T2-topology. In fact, we see from (2.5.11) that (T1) is fulfilled but (T 2) fails to bold, since the complement of any neighbourhood of x or y is finite and two sets of this kind cannot be disjoint.

2.5.

96

(2.5.24)

SEPARATION AXIOMS

2.5.e. Regular spaces. In the following axioms we shall speak about the separa· tion of a point and a closed set. (S 3) In the topological space [E, af] any point x EE and any closed set not con taining x are disconnected.

The S3-spaces are also called regular spaces. Axiom (S 3) is stronger than axiom (S 2): (2.5.24) Every regular space is an S2-space. Proof. If x and y are weakly separated, e.g. by (2.5.3) x a fortiori x and y are disconnected. I

~

y, then x and Ji and

Another important characterization of regular spaces is the following one: (2.5.25) A topological space is regular if! every point has a closed neighbourhood base (i.e. a neighbourhood base consisting of closed sets). Proof. If the space is regular and V is a neighbourhood of x, then x and E - int V are disconnected, say x E Vi, E - int V c V2 ,

Vi n V2 = 0

for suitable open sets Vi and V2 • Then by

Vi C

E - V2 = E - V2 C int V C V

J\ c

E - V2 c Vis a closed neighbourhood of x. Conversely if x ~ F = F, let V' be a closed neighbourhood of x contained in E - F. E - V' is then a neighbourhood of P-which does not intersect V'. I

On the pattern of (2.5.20), the following statement is valid: (2.5.26) Any subspace of a regular space is regular. Proof. According to (2.5.25), we have to observe that if, in the space [E, af], o(x) is a neighbourhood base of x consisting of closed sets and x E E0 c E, then by (2.3.10) and (2.3.13) o(x) (n) {E0 } is a (af I E0)-closed (af I E0)-neighbourhood base of x. I (2.5.27) If af; is a regular topology on E and 8T = sup {W;: i E J} then 8T is regular .as well. n Proof. Suppose that x E G and G is af-open. Then x E G;1 c G where G; 1

n 1

is a W;ropen set, i1 E I. If x E Hi; c Fi; c G;1 where Hi1 is afitopen, F';1 is afit n

closed, then x

n

n

En H; C n Fi C n Gi c 1

1

1

1

1

1

dosed. Thus we can apply (2.5.25).

n

G and here

I

From the union of (T0) and (S 3) we get: (T 3) The space [E, aT] is a regular T0-space.

n

n Hi is af-open, n Fi; is af1

1

1

(2.5.31)

2.5.e.

97

REGULAR SPACES

In view of (2.5.24) and (2.5.17) the following is true: (2.5;28) Every T 3-space is a T 2-space.

I

As a consequence of (2.5.26) and (2.5.6), we get: (2.5.29) Any subspace of a T 3-space is a T 3-space.

I

An example of a i:"2-space which is not regular is the following. Let E = R, and let the .neighbourhood filter of the points x -:f, 0, for the topology &r, be the same as for ~. while the sets ( - s, s) - A constitute a neighbourhood base for the point x = 0 where s > 0 and ·

We can see from (2.2.3) that Si is a topology finer than ~ and therefore by (2.5.22) is a T2-topology. It is obvious that the set A is &r-closed. However it is evident that O and A are not disconnected since if Vi E tJ(O), V2 E t>(A), then for suitable 8

>

0

(-s, s) - Ac suitable n EN and (-1- - J, - 1n n n necessarily Vi n V2 -:f, 0._

__!_ < s for

Vi,

+

DJ c V2 for some J > 0. Thus

2.5.f. Normal spaces. A topological space is said to be normal if two strongly separated sets are always disconnected; in other words if two·disjoint closed sets always have disjoint neighbourhoods. A normal space need not be regular even if it is a To-space. E.g. _the space [R, ~ J is a T0-space and is evidently normal,. since two closed sets in it can be disjoint only if one of them is empty. On the other hand, we know that j is not an Si-topology; thus it cannot be an S 3-topology either. This remark motivates the formulation of the following axiom:

(SJ The topological spa~e [E, Si] is a normal Si-space. (2.5.30) Every S4-space is an S 3-space. Proof. If x EF = F then by (2.5.8) x n F = and F are disconnected. I

0, thus x and F and a fortiori x

The normal spaces can also be characterized similarly to (2.5.25): (2.5.31) A topological space is normal if! every closed set F -:f, 0 has a closed neighbourhood base. Proof. If the space [E, Sf] is normal and F = F # 0, VE t>(F), then F c int v implies that the sets F and E - int V are. disjoint and closed. Suppose that Vi and V2 are open neighbourhoods of F and E -.- int V respectively su~h that 7 Akos Csaszar

2.5.

98

Vi n V2

(2.5.32)

SEPARATION AXIOMS

= 0. Then by Fc

Vi

C

Vi c E - V2 c int V c V

Vi is a closed neighbourhood of F and it is a subset of V. On the other hand, if F1 n F 2 = 0, Fi = Fi, F 2 = F2 , and Fi =I- 0, then E - F 2 is a neighbourhood of F 1 . If it contains a closed neighbourhood V of Fi, then 0. If Fi = 0 then 0 and E are the two disjoint neighbourhoods required. I

E - V:) F 2 is an open neighbourhood of F 2 and V n (E - V) =

On the analogy of (2.::i.26), we can now say: (2.5.32) The closed subspaces of a normal space are normal. Proof. Let"' [E, 8T] be a normal space, 0 =I- E 0 = E0 c E. If F 1 and F 2 are (8T I E 0)-closed, disjoint sets, then by (2.3.18) both are 8T-closed. Let G1 and G2 be 8T-open, Gin G2 = 0, F 1 c G1 , F 2 c G2 . Clearly G1 n E 0 and G2 n E0 are disjoint (8T I E0)-open neighbourhoods of the two given sets. I

Let us introduce the following axiom: (T4) The topological space [E, 8T] is an S 4-space and a T0-space.

We get then from (2.5.30): (2.5.33) Every T4 -space is a T 3-space.

I

We obtain from (2.5.32): (2.5.34) Any closed subspace of an S 4 -space is an S 4-space ,· any closed subspace of a T4-space is a T4-space. I

2.5.g. Completely normal spaces. A topological space is said to be completely normal if any two separated sets are disconnected. We get at once from the defini-

tion, because two strongly separated sets are separated: (2.5.35) Every completely normal space is normal.

I

On the other hand, a completely normal T0-space need not be regular, this being illustrated by the example of the space [R, 19]. It can be easily seen that two sets in it can be separated only if one of them is empty: a EA, b EB and e.g. a< b implies b E [a,+ m) c A, An B =I- 0. These facts motivate the formulation of the following axiom: (S 5) The topological space [E, 8T] is a completely normal Si-space.

From (2.5.35), we get at once: (2.5.36) Every S 5-space is an S 4-space.

I

The following characterization of completely normal spaces is a sharpening of (2.5.35) and can be compared with (2.5.32):

(2.5.40)

2.5.g,

COMPLETELY NORMAL SPACES

99

(2.5.37) The following statements are equivalent for a topological space [E, 8T]: (a) [E, Sf] is completely normal; (b) every subspace of [E, 8T] is normal,'. (c) every open subspace of [E, 8T] is normal. Proof. (a)=> (b): Let 0 -:f:. E0 c E, F1 n F 2 = 0, F1 and F 2 be (8f I E0)-closed. Then (denoting by a bar the 8f-closure as usual)

F1 n E0 = F 1 , F2 n E0 = F2, F1 n F2 = 0, i.e. F 1 and F 2 are separated with respect to 8f.

so that F1 n F 2 = Therefore there are 8f-open sets G1 and G2 such that F1

c

G1, F 2

c

G2 , G1 n .G2

= 0,

which implies that G1 n E0 and G2 n E0 are disjoint (8f I Eo)-neighbourhoods of F1 and F 2 . respectively. (b) => (c): obvious. (c) => (a): Suppose that A and B are 8f-separated. Then Ac

A - B = Fi, Be B -A= F 2,

F1 and F 2 are evidently disjoint; further both are closed in the subspace [E0 , 8T I E0 ] where E0

= E - (A n B) = (E - A) u (E - B).

In fact F 1 = A n Er,, F 2 = B n E0• E0 is 8f-open of course, so that F1 and F 2 can be included in disjoint (8f I E0)-open sets. These are of the form G1 n E0 and G2 n Eo, where G1 and G2 are 8f-open, hence they are themselves 8f-open. Finally AC G1 n E0 , B c G2 n E0 are the two disjoint 8f~open neighbourhoods we are looking for. I

In view of (2.3.17), we get immediately from (2.5.37): (2.5.38) Every subspace of a completely normal space is completely normal. Each subspace of an S 5~space is an S 5-space. I

Introducing the following axiom: (T5) The topological space [E, 8T] is an S 5-space and a T0~space, we can add: (2.5.39). Every T5 -space is a T4~space. Every subspace of a T5-space is a T5-

space.

I

Metric and pseudo-metric spaces give us important examples for T5- and S5spaces respectively: (2.5.40) Every metric space is a T5-space,· every pseudo-metric space is an Ss-

space. 7*

2.5.

100

SEPARATION AXIOMS

(2.5.41)

Proof. It is sufficient to show that every pseudo-metric space is normal. In fact, by (2.3.12) any subspace of a pseudo-metric space is pseudo-metric; hence we find from (2.5.37) that every pseudo-metric space is completely normal; thus by (2.5.15) it is an S5-space. Hence by (2.5.5) every metric space is a T5-space. Now if the sets F1 and F 2 are disjoint and closed in the pseudo-metric space [E, p], let G1

= {xEE:p(x,F1) O; thus F1 c G1 and similarly F 2 c G2 . Furthermore G1 is open; for, if x E G1 and B > 0 is so chosen that p(x, F 1)

then S

(x, ;) c

Gi, since y ES

+B
0 such that Vx = [x, x + B,b) does not intersect B. In the same way we can find for each x E B ap. Bx > 0 such that Vx defined as before does not intersect A. On account of (2.4.21) @+ I E0 is a Lindelof topology for the set Eo = U Vx u U Vx :::, A u B. xEA

xEB

Thus E0 can be covered with countably many Vx: 00

Eo

= U i=l

Vx; (xi EA u B, i EN).

(2.5.41)

2.5.h.

101

EXERCISES

We have to observe that the sets Vx are open and closed with respect to $+. Therefore the sets · n'-1

Gn

= Vxn - LJ Vx;•

G1

= Vx,

i=l

are evidently $+-open and disjoint, and none of them intersects A arid B simultaneously. Furthermore_ · 00

LJ

Gn = Eo ::J A u B.

1

Thus the union of the sets Gn intersecting A or B respectively furnish disjoint $+-open neighbourhoods of A. and B. I 2.5.h. Exercises. 1. Let F -:/- 0 be a closed set in the space [R, $]. Show that (a) if Fis bounded then it has a countable neighbourhood base; (b) if F = N, then it has no countable neighbourhood base.

[ In case (a), let Vn_ = {x: pi(x, F) < ~}. In case (b), for arbitrary open .

sets Gn

::J

.

1

N (n EN), let Xn EGn. be such that O < I Xn - n I < -, then en> 0 . 2

such that en < I Xn - n I, and G = .

U (n

- em n

nEN

·

+

en)·] ·

2. In the space [R, $] construct two sets which are (a) weakly separated but not separated; (b) disconnected but not strongly separated. 3. In the space [E, Ste], where Eis infinite, give an example of two sets which are strongly separated but not disconnected. 4. Show that the topological space defined in 2.2.g.4 is a T0-space without being an S1-space. 5. Suppose that the space [E, §"] figuring in 2.2.g.5' is a Ti-space consisting of more than one point. Show that the topology on ~(E) defined there is (T0) but is not (T1). 6. Give an example for topologies 8f1 and §"2 such that 8f1 < 8f2, 8l"1 is an S5 topology but 8l"2 is not an Srtopolcgy. [Let f 1 be the indisctete topology of R, 8l"2 = $.] 7. Give an example for topologies 8l"1 and 8f2 such that 8l"1 < 8l"2, 8l"1 is a T5 topology but f 2 is not regular. [Let 8l"2 be the non-regular topology defined in the example in 2.5.e and 8i1 = = &\.] 8. Let E be an infinite set, a EE, let ij consist of 0, E, and the finite subsets of E containing a. Prove that (a) SF is the system of closed sets of a topology 8i on E;

2.6.

102

__;

CONTINUOUS MAPPINGS

(b) Sf is normal ; (c) Sf IE - {a} is not normal. 9. Let Sf be the topology defined in the preceding exercise; Sf' is defined in the same way, giving the role of a to a point b:/=a. Show that gr and Sf' are normal but sup {Sf, Sf'} is not normal. 10. Let E be an· ordered set, Sf the order topology of E. The set Kc Eis said to be convex if a, b E K, a < x < b implies x E K. Prove that (a) if~ {i EI) is convex and ~ =I= 0 then U Ki is convex too;

n ~I

.

~I

(b) if a E A c E, then there exists a largest convex set containing a and contained in A; · . (c) Sf adinits a base consisting of convex sets; (d) if A is open then the convex set under (b) is open as -well. Suppose now that Mand N are Sf-separated, for x EM, let Kx be the largest convex set containing x and contained in E - N. Show that · (e) if Xi, x 2 EM; Kx; n Kx, ¥= 0, then Kx 1 = Kx,; (f). if y E N and V is a. convex neighbourhood of y ·not intersecting M, x 1 < < X2 < y, Xi, X2 EM, Kx, n V =I= 0, then Kx, = Kx,; . . (g) if y E N and V is a convex neighbourhood of y not intersecting M, then V intersects at most two of the sets Kx (x E M); (h) x EM, Yi EN, x A; if A C Y then J(f-1(A)) C A. I

(2.6.3) ff f: X

-+

.

Y is an injective mapping then, for any set A c X,

1- 1(f(A)) = (2.6.4) If A c f(X), in particular

A.

I

if f: X-+ Y is surjective and A c Y, then

J(f- 1(A)) = A.

I

(2.6.6)

2.6.b.

IMAGE AND INVERSE IMAGE OF A SYSTEM OF SETS

105

Suppose/: X---+ Y, X 0 c X, f(X0) c Y0 c Y. We give the name restriction to X 0 and Y0 of the mapping f, denoted by f I :f:, to the mapping g: X0 ---+ Y0 for which g(x) = f(x) whenever x EX 0 • When Y0 = Y we speak simply of the restriction off to X 0 and denote it by f I X 0 • We have already used a similar notation in connection with subspaces of metric (or pseudo-metric) spaces but not in a completely consequent manner as the metric (pseudo-metric) p can be considered as a mapping not from the underlying · set E but from the set of all pairs of elements of E. Suppose/: X---+ Y and g: Y---+ Z. We use the term composition of the mappings f and g for the mapping h: X---+ Z for which h(x) = g(f(x)) for x EX; it is denoted by the symbol h = go f We can see at once: (2.6.5) Suppose f: X ---+Y, g: Y---+ Z, h = gof (a) If Ac X then h(A) = g(f(A)); (b) if Ac Z then h- 1 (A) = J-1(g- 1(A)); (c) if f and g are injections, then h is an injection as well; (d) if f and g are Surjections, then h is a surjection as well; (e) if f and g are bijections, then his a bijection as well and h- 1 = 1- 1 0 g- 1 . (2.6.6)

If f:

I

X---+ Y, g: Y---+ Z, h: Z---+ U, then

ho(gof) = (hog)of; thus we can write quite simply h o go f without brackets. I The bijection /: X---+ X for which f(x) = x for arbitrary x EX is called the identical mapping or identity belonging to the set X. If X c Y and /: X ---+ Y is the mapping for which f(x) = x whenever x EX, then f is clearly an injection; it is called the canonical injection from X into Y.

2.6.b. Image and inverse image of a system of sets. Suppose/: X---+ Y and let us denote by ~(X) and ~(Y) the systems of sets consisting of all subsets of the sets X and Y respectively. If we assign to the set A c X the image f(A), then we · define a mapping /*: ~(X) ---+ ~(Y);

~e f*(A) = f(A/ for any set A c X, it does not lead to any misunderstanding if we drop the star and write simply f instead of/*. Similarly, by assigning to the set B c Y the set 1-\B) we define a mapping g: ~(Y)---+ ~(X) for which g(B) = 1- 1 (B) for arbitrary B c Y; if we write 1- 1 instead of g, it does not lead to any misunderstanding even if f is a bijection for in this case

whenever B c Y.

2.6.

106

CONTINUOUS MAPPINGS

(2.6. 7)

If mis a system of sets in X then as a logical extension of our agreements we shall denote by /(2:c) the system of sets {/(A): A E 2:c};

similarly, if ){3 is a system of sets in Y, 1- 1(){3) will denote the system of sets

{f- 1 (B): BE ){3}. From (2.6.1) we get at once (2.6. 7) Suppose f: X - Y and let 2X and ){3 be systems of sets in X. Then (a) m < ){) implies /(2:c) < /(){3); (b) m '"'-' ){) implies /(2:c) /(){3); (c) /(2:c (n) ){3) > /(2:c) (n)/(){3); (d) if t is a filter base in X, then f(r) is a filter base in Y; r-_;

(e) if c is a centred system in X then f( c) is a centred system in Y; (f) /(2:c (u) ){3) = /(2:c) (u)/(){3). I

(2.6.8) Let f: X - Y be a surjection. Then (a) if 2X is an ascending system in X, then f(m) is an ascending system in Y; (b) if ){3 is the ascending system in X generated by the system m, then J(){3) is the ascending system in Y generated by f(l)X); (c) if & is a filter in X, then/(&) is a filter in Y; (d) if t is a filter base in X and & is the filter in X generated by it, then f(&) is the filter in Y generated by f(r). Proof. (a) If A Em and f(A) c B c Y 1 then by (2.6.4) B = f(f- 1 (B)), and by (2.6.2) A C / - 1(f(A)) C f- 1(B) C X, so that J- 1(B) E mand B Ef(m). (b) From this and from (2.6.7) it follows that if ){3 is an ascending system in X equivaleiit to m then /(){3) is an ascending system in Y equivalent to /(I)!). (c) is a consequence of (a) and (2.6.7) (d), while (d) results from (c) and Cb). I We get easily from (2.6.2), (2.6.3), and (2.6.4): (2.6.9) Suppose f: X - Y and let 2X and ){3 be systems of sets in Y. Then (a) if m < ){3 then 1- 1 (2:c) < 1-1(){3); (b) if m,. ___, ){) then 1- 1(1):() ,. ___, 1- 1(){3); (c) 1- 1 (u) ){3) = 1-1em) (u)J- 1 (){3); (d) 1-1 (n) ){3) = 1- 1 (2:() (n)J- 1 (){3); (e) if r is a filter base in Y and 0 ~ t (n) {f(X) }, in particular if f is a surjection, then J- 1 (r) is a filter base in X; (f) if c is a centred system in .Y and f is a surjection, then 1- 1 ( c) is a centred system in X; (g) J(f- 1(2:()) > mand equality holds here if f is a surjection;

cm cm

(2.6.12)

2.6.c.

CONTINUOUS MAPPINGS

107

(h) for any system G£ of sets in X f-l(f(G£)) < (£, and equality holds here

if f is an injection.

I

2.6.c. Continuous mappings. We say that the function of one variable/: R - t R is continuous at x 0 E R if, for every e > 0, there is a 8 > 0 such that I x - x 0 I < 8 implies I f(x) - f(x0) I < e. The condition occurring here can be formulated as follows: for every spherical neighbourhood S(f(x0 ), e) of the point f(x0 ) E R, there must be a spherical neighbourhood S(x0 , 8) of the point x 0 E R such that x E S(x0 , 8) implies f(x) E S(f(x0 ), e). In a similar way, the function of m variables f: Rm - t R is continuous at x 0 E Rm if, for each e > 0, there is a 8 > 0 such that p(x, x0 ) < 8 implies If(x) - f(x 0 ) I < e, i.e. if, for each spherical neighbourhood S(f(x0 ), e) of f(x 0 ) ER, there is a spherical neighbourhood S(x0 , 8) of x 0 E Rm with the property that x E S(x0 , 8) implies f(x) E S(f(x0 ), e). Hence the following definition is plausible. Let [X, °\9] and [Y, CW] be two neighbourhood spaces, f: X - t Y, and denote the °\9-neighbourhood filter of x EX by b(x), and the CW-neighbourhood filter of y E Y by tu(y). We say that the mapping f is continuous at the point x 0 E X if we can find for each neighbourhood WE tu(f(x0)) a neighbourhood VE b(x0 ) such that x E V implies f(x) E W, briefly f(V) c W.

(2.6.10) Let [X, "\9] and [Y, CW] be two neighbourhood spaces with the neighbourhoodfilters b(x) andtu(y) respectively (x EX, y E Y),further f: X - t Y. The mapping f is continuous at the point x 0 E X if! (2.6.11) or equivalently

f(b(x0)) > tu(f(x0)),

~ff

(2.6.12) In the conditions (2.6.11) and (2.6.12), we can replace b(x0) and tu(f(x0)) by an arbitrary °\9-neighbourhood base b(x0 ) of x 0 and a CW-neighbourhood base b'(f(x0 )) of f(x 0) respectively. Proof. (2.6.11) as well as (2.6.12) exactly express the condition figuring in

the definition of continuity. (2.6. 7) (b) and (2.6.9) (b) show that b(x0) or tu(f(x0)) can be replaced by any equivalent filter base. I We have to notice that since 0 ~ tu(f(x0)) (n) {J(X)} the system of sets f- 1 (tu(/(x0))) figuring in (2.6.12) is a filter base according to (2.6.9) (e). Our theorem shows that, by considering e.g. a mapping from a pseudo-metric space into a pseudo-metric space, spherical neighbourhoods can be used in the definition of continuity; in particular, we get the notion of continuity known from elementary analysis in the case [X, "\9] = [Rm, &\m], [Y, CW] = [R, &\]. On the

108

2.6.

CONTINUOUS MAPPINGS

(2.6.13)

other hand, by considering a maptJing from a topological space into another topological space, we can speak e.g. of open neighbourhoods in the conditions defining the continuity. If it is important to emphasize the neighbourhood structures ~ and ~ figuring in the definition of continuity, then we say that the mapping/is (~, ~)-continuous at the point x0 • , It is not difficult to see that a mapping/: R-+ R is (@+, @)-continuous ((@-, @)continuous) at the point x0 ER iff the real function f of one variable is continuous on the right (on the left) at this place. The function /: Rm -+ R of m variables is said to be semi-continuous from above at x0 ERm if, for every e > 0, there is a 0 such that x ES(x0 , b) implies f(x) tJ(x0), thus by (2.6.7) /(t) > f(tJ(x0)), and by (2.6.11) f(tJ(x0)) > tu(f(x0)), so that /(t)-+ f(x0). Conversely tJ(x0) -+ x 0 implies /(tJ(x0)) -+ f(x0), hence

/(tJ(x0)) > tu(/(x0)) which ensures (2.6.11) i.e. the continuity off

I

As a more exact generalization of the quoted theorem: (2,6.14) Let [X, 3T] be an Mi-space, [Y, ~] an arbitrary neighbourhood space, X-+ Y, x0 EX. f is continuous at the point x0 if! f(xn) -+ f(x 0) for any .sequence (xn) of points of X converging to x0 • · Proof. The necessity of the condition follows, even in the case of an arbitrary neighbourhood space [X, ~], from (2.6.13) for if t denotes the sequential filter

f:

base belonging to (xn), thenf(t) is evidently the sequential filter base belonging to the sequence (f(xn)). Suppose now that f is not continuous at the point x0 • Then f(x0) has a neighbourhood W such that f(V) c W does not hold for any neighbourhood V of x0• We may on account of (2.4.13) take a neighbourhood base {Vn: n EN} of x 0 such that Vi ::, V2 ::, V3 ::, ••• , and let Xn E Vn be such that f(xn) ~ W. Evidently Xn -+ x 0 but f(xn) does not converge to f(x 0). I The well-known theorem concerning the continuity of the composition of functions can be generalized as follows:

(2.6.20)

2.6.c,

109

CONTINUOUS MAPPINGS

(2.6.15) Let [X, ~i], [Y, ~ 2 ], [Z, ~ 3 ] be three neighbourhood spaces, f: X--, Y, g: Y _, Z, h = gof, x 0 EX, Yo= f(x 0 ). If f is continuous at the point x 0 and g is continuous at the point y0 , then h is· continuous at the point x 0 • Proof. By (2.6.5) (a) and (2.6.7) (a) h(bi(x0)) = g(j(b 1 (x0))) > g(bly0)) > > ba(g(yo)) = bih(xo)). I

We say that the mapping f: X--, Y, where [X, ~] and [Y, CW] are again two neighbourhood spaces, is continuous (or (~, CW)-continuous) if it is continuous at every point x0 E X. E.g. if f(x) = Yo E Y for each x EX, then f is clearly (~, CW)-continuous as f(b(x0)) = {{Yo}} > ttJ(y0). Briefly, every constant mapping is continuous. In a similar way: (2.6.16) If [X, ~] and [Y, CW] are two neighbourhood spaces, gr is the indiscrete topology on Y and f: X--, Y is arbitrary, then f is (~, 8J)-continuous. I

(2. 6.17) Let ~ and CW be two neighbourhood structures on X. The identical mapping f of the set Xis (~, CW)-continuous if[~> CW. Proof. x 0 E X implies f(b(x0)) = b(x0). I (2.6.18) Let f: X _, Y be(~, CW)-continuous at the point x 0, ~ 1 be a neighbourhood structure on X finer than ~. and CW1 a neighbourhood structure on Y coarser than CW. Then f is (~i, CWJ-continuous at the point x 0 • · Proof. The statement results from (2.6.15) and (2.6.17) for if g and h denote the identities of X and Y respectively, then clearly f = ho fog. I

E.g. a continuous function of one variable is continuous from the right and from the left since $ < $+, $ < i-. (2.6.19) Let f: X--, Y, ~ be a neighbourhood structure on X, gr; (i EI) a topology on Y, gr = sup {gr;: i EI}, x 0 EX. f is (~, 8J)-coniinuous at the point x 0 if[ it is (~, gr;)-continuous therefor each i EI. Proof. From the (~, gr)-continuity there follows the (~, 8l;)-continuity for each n

ion account of (2.6.18). On the other hand, by (2.3.3) the sets of the form

n W;k, k=l

where W;k is a 8l;k-open set containingf(x0) and ik EI (k = 1, ... , n), constitute a 8J-neighbourhood base of/(x0). If/is(~, gr;)-continuous at the point x 0 for all i, then we can find Vik E b(x0) for which !(Vik) c Wik (k = l, ... , n). For the set n

V

=

n

n Vik E b(xo), we have f(V) C k=l n W;k. I k=l

E.g. it can be seen from the equality sup {i, ~} = @, that a function f: Rm --, R is continuous iff it is semi~continuous both from above and from below. -,~

(2.6.20) Let [X, ~] be a neighbourhood space, 0 # X 0 c X. The canonical injection from X0 into Xis (~ I X0 , ~)-continuous. Proof. x 0 E X 0 implies f(b(x 0 ) (n) {X 0 }) = b(x0) (n) {X0 } > b(x0). I

110

2.6.

(2.6.21)

CONTINUOUS MAPPINGS

(2.6.21) Suppose f: X-+ Y, and let [X, ~] and [Y, CW] be two neighbourhood spaces, f(X) c Y0 c Y, x 0 E X, g = f [ i'. The mapping f is (~, CW)-continuous at the point x0 if! g is (~, CW I Y0 )-continuous at the same place. Proof. f(b(x0)) = g(b(x0)) is a filter base in Y0 , thus it is simultaneously finer than ttJ(f(x0)) or than ttJ(g(x0)) (n) {Y0 }. I (2.6.22) Suppose f: X -+ Y, x 0 EX0 C X. If f is (~, CW)-continuous at the point x 0 , then fl X 0 = g is (~ I X0 , CW)-continuous at the same place. Proof. g(b(x0) (n) {X0 }) = f(b(x0) (n) {X0 }) > f(b(x0)) > ro(f(x0)) = ttJ(g(x0)). I (2.6.23) Let [X, 8fi] and [Y, 8f2 ] be two topological spaces, f: X-+ Y, ®2 a 8f2subbase. The following statements are equivalent: (a) f is (§"i, 8f2 )-continuous; (b) the inverse image of every §°2-open set is 8rropen; (c) the inverse image of every set SE ®2 is 8f1-open; (d) the inverse image of every §°2 -closed set is 8rrclosed; (e) Ac X implies f(A) cf(A). Proof. (a)==> (b): If G is 8f2-open and x Ef- 1 (G), then G is a 8f2-neighbourhood of f(x), thus x has a 8lrneighbourhood V for which Vcf-1(G). Hence every point of f- 1 (G) is an interior point. (b) ==> (a): if x EX and o(f(x)) is any open neighbourhood base of f(x), then f- 1(o(f(x))) consists of open sets containing x and it is therefore coarser than the neighbourhood filter of x, thus (2.6.12) is fulfilled. (b) ==> (c): obvious. n;

(c) ==> (b): On account of (2.6.2) (b) and (c), G =

U B;, B; = iEI

n Sii,

Su E ®2

j=l

implies

f- 1 (G) = U f- 1(B;), iEI nt

r· (e): f-1(f(A)) is a closed set containing A, thus A cf- 1(f(A)) (f(A) denotes of course the 8f2-closure, A the 8f1 -closure) and f(A) cf(A). (e) ==> (d): If Fe Y and F = F, then, with the notation A =f- 1(F),f(A) c c f(A) c F = F, i.e. A c f- 1 (F) = A and A is closed. I E.g. the function f: Rm -+ R is semi-continuous from above (from below) iff, for all c ER, either f- 1( ( - oo, c)) is open or f- 1 ([c, + oo)) is closed (either f-1((c, + oo )) is open, or f- 1(( - oo, c]) is closed). f is continuous iff, for all c ER, f-1(( - oo, c)) and f- 1((c, + oo )) are open since the sets ( - oo, c) and (c, + oo) constitute a subbase for ©.

(2.6.26)

Y,

2.6.d.

HOMEOMORPHY

111

(2.6.24) Let ST; (i E[)be a topology on X, 8T = inf{Sf;: i E/}, Sf' a topology on f: X --+ Y. f is (ST, 8T') 0 continuous if! it is (§;, §')-continuous for all i. Proof. J-1(G) is Sf-open for a Sf'-open set Giff it is Sf;-open for all i, I

--E.g. f: R --+ R is continuous iff it is continuous both· on the· right and on the left, because $ =inf{$+,$-}. 2.6.d. Homeomorphy. Let [X, 8f1 ] and [Y, 8f2] be two topological spaces. The mapping f: X-+ Y is said to be a homeomorphism (more exactly a (Sfi, § 2)homeomorphism) if it is bijective, (Sfi, Sf2)-continuous and 1- 1 is (Sf2, § 1)-continuous.

(2.6.25) lffis a (8Ti, 8T2 )-homeomorphism, thenf- 1 is a (§2 ,STJ-homeomorphism. If f is a (8Ti, 8T2)-homeomorphism, g a (8f2, 8T3 )-homeomorphism, then gof is a (8Ti, 8T3 )-homeomorphism. Proof. (2.6.5) and (2.6.15), I

If there exists a homeomorphism which maps the topological space [X, § 1 ] onto the topological space [Y, 8f2], then we say that the two spaces are homeomorphic. Homeomorphy is clearly a reflexive .(identity is a homeomorphism), by (2.6.25) symmetric and transitive relation. By (2.6.23) f is a (Sfi, STJ-homeomorphism iff it transforms the system of § 1 open sets into the system of Sf2-open sets, or the system of 8f1 -closed sets into the family of 8f2-closed sets. As all notions defined in topological spaces (neighbourhood, interior point, convergence, etc.) can be reduced to the determination of open sets, so all those topological statements which can be made about a topological space (e.g. that it is an M 1-space, an M 2-space, that it fulfils some separation axiom, it is a Lindelof space, it is separable, etc.) will also be true for all spaces homeomorphic with it. Therefore the topological spaces homeomorphic with each other can be considered equivalent from a topological point of.view. For this reason the term topological mapping can be also used instead of the term homeomorphism and those properties of topological spaces which remain invariant under such mappings are called topological invariants. According to the foregoing all properties whose existence can be controlled as soon as the open sets are known are of this kind. We say that a topological space [X, Sfi] can be topologically embedded into the topological space [Y, Sf2] if [X, 8f1 ] is homeomorphic with a subspace of [Y, 8f2 ]. A bijection f: X--+ Y0 c Y which is (Sfi, 8f2 I Y0)-continuous and whose inverse is (8f2 I Y0, 8fJ-continuous is called a topological embedding of the space [X, Sfi] into [Y, Sf2]. The topological space [Y, 8f2 ] is said to be a continuous image of the topological.space [X, Sfi] if there exists a (Sfi, 8f2)-continuous surjective map f: X--+ Y. Some properties of topological spaces go over to continuous images of the space. E.g. (2.6.26) A continuous image ofa separable space is again separable.

2.6.

112

(2.6.27)

CONTINUOUS MAPPINGS

Proof. Let S be a dense countable set in the topological space [X, 8fi],f: X--+ Y a surjection, 8f2 a topology on Y and f (sr1 ,. 8i"2)-continuous. By (2.6:23) (e) Y = = f(X) = f(S) cf(S), thus f(S) is sr2-dense and clearly countable. I· 2.6.e. Continuous functions. In what follows we shall always understand by a function a real function, i.e. a map/: X--+ R where X is an arbitrary set. If [X, "\91] is a neighbourhood space, then the function f defined ort Xis said to be continuous at the point x0 EX if it is (~, $)-continuous there, and it is said to be continuous if this holds for every point x0 E X. As a generalization of theorems well known in elementary analysis, we have: (2.6.27) L;t [X, "\91] be a neighbourhood space, f and g functions given on X,

x0 EX, and suppose that f and g are continuous at the point x0 • Then the following functions are continuous here: (a) f + g; (b) Jg; (c) max(!, g); (d) min (f, g); (e) f if xE X implies g(x) =/: 0. g Proof. Let B > 0 be given, 1'J > 0 arbitrary for a moment, Vi and V2 neighbourhoods of x0 such that x E Vi implies lf(x) - f(x0) I < 1'/, x E V2 ini.plfos I g(x) - g(x0) I < 1'/, V = Vi n V2, and let us use the notations

u = f(x), ·Uo = f(x0), v = g(x), v0 = g(x0 ). (a) With the choice 1'/ =

I (u + v)

- (uo

; , x EV implies

+ Vo) I ~ I u - Uo I + I v

--: Vo

·.

(b) For Uo-:;, 0, v0

I< 1'/ + 1'/ = B• .

. .-:;,

. ll

8

8

0, let rt be chosen smaller t h a n ~ · ~ - ,

3, _

and if u0 = 0 or v0 = 0, let us drop the fraction with denominator 0. For x E V, we have

I uv

- UoVo I ~ I Uo I . .

· IV

Vo I

.

·

+ I VoJ • I u .

2

ll

Uo

I + Iu ll .

- Uo I · I V

-

Vo I


con-

(2.6.28)

2.6.e.

113

CONTINUOUS FUNCTIONS

min(/, g) =

f: I!; I· g

+

g

This follows by (2.6.15) from the fact that the map h: R - R defined by h(u) = = I u I is continuous since·

{~~ c, c),

h-1(( - co; c)) = h_ 1 ((c,

+ oo.))

if if

=·{R(- oo, -c) u (c, + oo), 0

C

> 0,

C

;:i; 0,

if c ~ 0, if C < 0,

thus (2.6.23) (c) can be used. (e): In view of (b ), it is sufficient to show that

2_

is continuous at x 0 , thus g by (2.6.15) and (2.6.21) that the map k: (R - {O})--+ R defined by the formula k(u) =

2_ u

is (&i JR - {O}, @)-continuous, since

I(l(; , I

k- 1((c,

+ (X) )) =

g

kog

if

C

> 0,

if

C

= 0,

o),

if

C

< 0,

(o, ; ),

if

C

> 0,

+ (X) ),

if

C

= 0,

oo, ; ) u (0, + oo ),

if

C

< 0,

oo, 0) u (; ,

k- 1 (( - co, c)) =

+ oo),

2_ =

( - oo, 0),

(0,

l(-

and we can refer again to (2.6.23).

\!-(0}.

However

I

In order to generalize a further important theorem of analysis we have to recall the fact that a sequence (f,,) of functions defined on X converges uniformly to the limit function/if, for each a> 0, there is an n0 EN such that n ~ n0 implies If,,(x) - f(x) I < e for each x EX. (2.6.28) If in the neighbourhood space [X, ~] the sequence (f,,) of functions converges uniformly to f and at a point x 0 E X all functions f,, are continuous, then f is also continuous there. Proof. For a> 0, let us choose an index n0 EN such that n ~ n0 implies

If,,(x)

.

- J(x) I
8l'i, if A83'2 B implies A83'1 B. According to (2.3.1): (3.1.17) Let 83'1 , 83'2 , 83' 3 be proximities on E. Then (a) 81'1 < 81'1; (b) 81'1 < 81'2 and 81'2 < 81'1 imply 81'1 = 81'2; (c) 81'1 < 81'2, 81' 2 < 81'3 implies 81'1 < 83'3. I The following theorem gives a connection between the comparison of topologies and proximity relations: (3.1.18) If 83'1 and 83'2 are proximities on E and 83'1 < 83'2 , then 1,)i(A) < 1JlA) for any set A c E. Therefore 8T@, < 8T@,·

3.1.

126

(3.1.19)

PROXIMITY SPACES

Proof. P E ,)Ji(A) implies A ~ E - P and therefore A &2 E - P, P E +ilA). The second statement is obtained from this by choosing A = {x}. I It is important to remark that, on the other hand, &f81 < &f&, does not imply &1 < &2 in general. For this purpose it is sufficient to find proximities &1 # &2 for which W&, = &f&, ff now E = R, p(x, y) = I x - y I and &1 = §'p, while &2 is the proximity defined according to (3.1.8) with the help of the topology &fP = $, which is (T5 ) by (2.5.40) and therefore a fortiori (S4), then, according to (3.1.12) and (3.1.13), W&, = &f&, = E, although &1 # &2, since e.g. if A = N, 1

B = { n + 2~ : n

EN},

then p(A, B~ = 0 implies A &1 B; on the other hand

A= A, B = ii, AnB = 0 implies A &2 B. It should be observed that in this example &1 < &2 , This can be seen from the

following theorem: (3.1.19) Let [E,&f] be a normal space, & the proximify defined in (3.1.8), &' an arbitrary proximity on E for which &f&, < &f. Then &' < &. Proof. If A §' B, i.e. A n Ji =I 0 where we denote by a bar the closure with respect to &f, then A&' Bas otherwise by (3.1.14) the 8!81 ,-closures of A and B would be &'-far from each other, and by (P 2) they would not intersect each other; hence by (2.3.2) {d) An Ji = 0 would hold a fortiori. I

On account of (3.1.13), we get from (3.1.19): (3.1.20) If [E, &f] is an S4-space, then & defined according to (3.1.8) is the finest proximity inducing 8!. I It is obvious that there exist on any set Ea coarsest and a finest proximity; with respect to the first, any two non-emI)tY sets are near, with respect to the latter, only the sets intersecting each other are near. The former induces the indiscrete topology of E, the latter the discrete topology of E, thus they can be callee the indiscrete and discrete proximity of the set E respectively. Similarly to (2.3.3), we can speak of the supremum and infimum of proximities. In fact:

(3.1.21) Let &i (i E J # 0) be proximities on E. Suppose that A & B holds for A, B c E if! there are, for any finite decomposition p

(3.1.22)

A= LJA·1' 1

q

B =!LJBk ):. ' 1

j and k such that Aj &i Bk for every index i E I. & is then the coarsest of the proximities finer than all &; and is denoted by

& = sup {&i: iE/}.

(3.1.22)

3.1.d.

COMPARISON OF PROXIMITY RELATIONS

127

For the corresponding topologies we have:

8f1 = sup {8f, 1: i EI}. Proof. It is obvious that & satisfies (Pi) and (P4). If A n B =f. 0, say x EA n B, then for any decomposition (3.1.22) and for suitable j and k, we have x E Ai n Bk and then, for any i E J, Ai &i Bk; thus SJ fulfils (P2). If A & B, Ac A', B c B', and · q

p

.B' = LJ B£

A. '-UA' -. j,

1

1

then p

A. =

q

LJ (A1n A),

B

= LJ (B£ n B),

1

1

thus A1n A SY; B£ n B for suitable j, k and for every index i E J, and a fortiori i Cand Bi C then there exist decom1 positions A &; B£; accordingly & satisfies (P 3). If A

q

p

C = LJCk,

A= LJAi, 1 p'

1 q'

C =

B=UB,, 1

LJ c:,, 1

such that for every j, k, l, m and suitable i = i(j, k) and i' = i'(l, m) Ai &; Ck, B1 &;,

c:,,.

Then taking arbitrarily one member from each of the decompositions p

= LJ

A uB

p'

Ai u

i=l q q'

C=

LJ LJ

LJ

Bi,

l=l

(Ck n

c:,.)

k=l m=l

for suitable i E I and i' E1 Ai &; Ck n

c:,.

and B1 &i, Ck n c:,,.

Thus (P5) is valid for &; . Finally if A SJ B, then there exists a decomposition of the type (3.1.22) such that Ai Sfi Bk for arbitrary members Ai and Bk and for a suitable index i = i(j, k). Let f'.ik and Qik be sets for which f'.ik n Qik = 0 and Ai &,u,k) E - I'.ik'

Bk &iu, k) E - Qik

further p

I'.i

==

n I'.ik ,

k=l

q

p

=

u I'.i,

i=l

1

.3.1.

128

(3.1.23)

PROXIMITY SPACES

Clearly Pi n Qi = 0 for all j, so that P n Q = 0. Moreover it is evident that ifC &; D at least for one i EJ, then Ci D so that, for all j and k,

and using (P1)-(P5) already proved

and a fortiori -

Ai &

p

n (E i=l

p

LJ

Pi) = E -

Pi = E - P

i=l

and p

LJ

A=

Ai8YE- P.

i=l

In a similar way

thus

and finally q

-

B = LJBk@E- Q. 1

Accordingly & fulfils (P6) too. Therefore SJ',is a proximity on E. We mentioned already that if C ii D for some i, then C & D, i.e. &i < &. On the other hand, if @i < &' for a proximity 8Y' for every i EJ, and A SJ B, then there exists a decomposition of the form (3.1.22) for which Ai ~i(i,k) Bk for every j and k. Then Ai&' Bk for a-,1 j and k, so that p

A= UAj&' Bk, 1

-

q

A &'LJBk

= B,

1

i.e. 8l' < &'. According to (3.1.18) Sf811 < Sf81 for every i, hence Sf < 8'81 with the notation

Sf= sup {8f811 : iEJ}. On the other hand, if Vis a Sf81,neighbourhood of x, i.e. if {x} q

E- V=UBic, 1

iE-

V, then

( 3.1,27)

and {x}

3.1.e. &;Ck) .

RESTRICTION OF PROXIMITIES

129

EI. Therefore

E - Bk is a 8f'81i(k>:-neigh-

Bk for a suitable index i(k)

bourhood and afortioti a 8r'-neighbourhood of x, so th:at V = · is a 8f'-neighbourhood, i.e. 8f'81 < 8f'. I

q

.

n(E ~ Bk)

too

1

(3.1.23) Let &; (i EJ ¥= 0) be a proximity on E. There exists a proximity & which is the finest of all proximities coarser than all &;, it is denoted by & = inf {$;: i EI} . '

Proof. Since the indiscrete proximity of the set Eis coarser than all &1, we can speak of the supremum .of the proximities coarser than all &1• Denoting it by &, we clearly obtain a relation with the required property. I

(3.1.24) If a topology 8r' can be induced by a proximity, then there exists a finest , one among the proximities ,inducing 8r'. It is qalled the Cech-Stone proximity of 8r'. Proof. On account of (3.1.21), the supremum of all proximities which induce 8f' induce_s 8f' as well. I 3.1.e. Restriction of proximities. ,Let p be a pseudo•metric on E, 0 ¥= E0 c E, and consider the pseudo-metric spaces obtained from the pseudo-metric p on E and E0 respectively. If A, B c E0 , then the validity of p(A, B) = 0 does not depend on whether we consider A and Bas subsets. ofE or E0• Therefore the-following theorem and the definition contained in it are, plausible:

(3.1.25) Let & be a proximity on E, 0 ¥= E0 c E. For two subsets A, B c E0 , let A, &0 B hold iJf A & B. &0 is then a proximity on E0 called the restriction to E0 of the proximity .& ;· in symbols: &0 = & I E0 • Proof. (PJ-(P5 ) are evidently fulfilled by &0 • lf A &0 B, i.e. A, B c:: E 0 : and A @B, then there are P and Q such that P n Q = 0, A 1& E - P~ B & E -'- Q. Let P0 = P nE0 , ·Q0 = Q ri E0 • Evidently P0 n Q0 = 0, then E0 - P0 CE - P E0 ~ Q0 C E - Q implies· A & E0 - P0 , B j E0 · - Q0 , thus A &0 E0 ..:.. P0 :

B~&-~I

. ·

We understand by the subspace of the proximity space [E, &] •belonging to the set 0 ¥= E0 CE the proximity space [E0, & I E0 ]. Our introductory remark can be formulated as follows: (3.1.26) The proximity of a subspace of a pseudo-metric space is the restriction of the proximity of the whole space. 1· · · · · (3.1.27) If [E, &] is a proximity space·, ·0 ¥= Eo t:: E, then the Si' I E~-proximity filter of any set 0 ¥= Ac E0 is .))(A) (n) {E0 }. Therefore 8r',1s,

= 8r'sr I Eo.

Proof. If P E .))(A), i.e.. A_ 8F E - ,P, then E0 -: (P n E0) c E - P implies A&0 E0 - (P n E0 ), and. P n E0 E ,P 0 (A), where &0 = 8F J,E0 and .):1 0(A) denotes the 9 Akos Csaszar

3.1,

130

(3.1.28)

PROXIMITY SPACES

&0-proximity filter of A. Conversely if P0

E ,\J 0 (A), i.e. P0 C E0 and A &0 E0

P0 , then, with the notation P = P0 u (E - E0), E - P = E 0 - P0 implies A & E - P, P E ,\)(A) and P0 = P n E 0 • The second statement follows from this applied for the special case A = {x}. I -

Valid on the pattern of (2.3.15) is: (3.1.28)

If &1 and &2 are proximities on E, 0 # E0 c E, and 83'1 < 83'2, ihen &1 I Eo < &2 I Eo · I

The following corresponds to (2.3.16) and is obtained immediately from (3.1.21): (3.1.29)

If 83'; is a proximity on E (i E/), & =sup{&;: i EI}, 0 sup{&; I E0 : i EI} = & I Eo.

# E 0 CE, then

I

Finally, similarly to (2.3.17): (3.1.30)

If 83' is a proximity on E, 0 # E0 C E 1 CE, then (& I E1) I Eo

=

& I Eo ·

I

3.1.f. Inverse image of proximities. Similarly to the case of topologies, the restriction of proximities can be considered as the special case of a more general concept:

(3.1.31) Let f: X ~ Y, & be a proximity on Y. For the sets A, B c X, let A &0 B hold if.! f(A) & f(B). Then ©'0 is a proximity on X called the inverse image of the proximity SJ' and denoted by 1-1(&). Proof. (P 1)-(P5 ) obviously hold for &0. If A &0 B, i.e. f(A) &f(B), then there 1(P) n are P and Q such that P n Q = 0, f(A) SY Y - P, f(B) sf Y - Q. Then nJ-1(Q) = 0, A 8Yo X -J-1(P) = f-1(y - P), and Bio X - J-1(Q) = J- 1(Y -Q) as J(f-1(Y - P)) c Y - P, J(f-1(Y - Q)) c Y - Q implies f(A) 'ifJ{f- 1(Y - P)), f(B) SYJ(f-1(Y - Q)). Thus (P 6) holds. I

.r-

From (3.1.25) and (3.1.31) we get immediately: (3.1.32) If [E, &] is a proximity space, 0 # E 0 CE, f: E 0 ~Eis the canonical injection, then 1- 1 (&) = & I E0 • I As a generalization of one part of (3.1.27), we get: (3.1.33)

If f: X

~

Y, and SJ' is a proximity on Y, then

1-1(&I,) =

&I

-1 (/§).

Proof. By (2.6.35) the sets J- 1(P) where PE ,\J( {J(x) }), i.e. {f(x)} & Y - P, constitute an 1-1(&I,)-neighbourhood base of x EX. However, for such P evidently

3.1.f.

(3.1.38)

INVERSE IMAGE OF PROXIMITIES

131

{x }J- 1 (8l')J- 1(Y - P), as J(f- 1(Y - P)) c Y - P implies {f(x)} &J(f- 1(Y - P)), so that X - J- 1(Y - P) = 1-1(P) is a 8i1 -,{&J-neighbourhood of x. On the other hand, if V is a 811 -,c&J-neighbourhood of x, then {f(x)} i f(X - V) implies P = = Y - f(X - V) E 1J( {f(x))} and clearly J-1(&) c V. I (3.1.34) lff: X--+ Y, &1 and &2 are proximities on Y and &1 < Si' 2, thenf- 1(&1) < q(f(A)), B c Y implies

1-1(q(B)). < ,p(f-1(B)) . Therefore f is (ST@, STe,)-continuous. Proof. If Q E q(f(A)), i.e. f(A) a_ Y - Q, then by (3.1.38)

A

cf-1(f(A)) °iJ- 1(Y - Q),

i.e. P = X - J- 1(Y - Q) E ~(A), f(P) Ef(,p(A)), and obviously f(P) c Q. If P 1 EJ-1( q(B)), i.e. P 1 = J-1(Q1), Q1 E q(B), then B0. Y - Q1 .implies @J- 1(Y - Q1), i.e. P1 = X - J-1(Y - Q1) E ,p(f-1(B)). In particular, by choosing A = {x},

.r-1(B)

1

/(,p({x})) > q({f(x)}),

so that by (2.6.10) / is (ST@, STe,)-continuous.

I

On the pattern of (2.6.15), it is true that: (3.1.41) If f: X-+ Y is (&i, &2 )-proximally continuous and g: Y-+ Z is (&2 , &3)proximally continuous, then h = g of is (&i, &3)-proximally con.tinuous. I The following statement corresponds to (2.6.17): (3.1.42) Let & and Q be two proximities on X. The identity f of the set Xis (&, 0)proximally continuous if! & > 0. I

(3.1.49)

3.1.g.

PROXIMALLY CONTINUOUS MAPS

133

We get from (3.L42) and (3.1.41) on the pattern of (2.6.18): (3.1.43) Let f: X-+ Y be (@, el)-proximally continuous, @1 a proximity on X finer than@, el1 a proximity on Y coarser than el. Then/is (@i, el1 )-proximallycontinuous. I

Corresponding

fo

(2.,6.19), we can say:

(3.1.44) Let f: X -+ Y, '@ be a proximity on X, el; (iE I) proximities on Y, Q = sup {el;: i E I}. f is (@, el)-proximally continuous iff it is (@, el;)-proximally continuous for each i. Proof. According to (3.1.39) /is (@, el)-proximally continuous iffJ-1(6,) < @, and it is (@, Ll;)-proximally continuous iffj"'. .\Q;) < @. Thus the statement follows from (3.1.35). I

Corresponding to (2.6.24) we have: (3 .1.45) Let f: X -+ Y, @;(i E /) be proximities on X, @= inf {@;: i E I}, Q a proximity on Y. f is (@, Q)-proximally continuous if! it is (&;, elfproxiinally continuous for each i. Proof. Let@' = 1-1 (Q). If/is(&;, Q)--proximally continuous for each i, then by (3.1.39) @' is a (8J', ~ lf(X))-equimorphism; (c) if f is bijective, then 8l' = 1-1(~) holds iff f is a (&, ~)-equimorphism, I 3.1.h. Exercises. L Let [E, p] be a pseudo-metric space and, for A, B c E, let A & B hold iffat least one of A and Bis bounded and An B = 0. Show that 8l'

,~

.,.1roximity. [If A is bounded and An B # 0, then there is an open P such that Ac Pc c Pc E - Band if x EP, then p(x, A)< l.] 2. Let [E, Sf] be a topological space and SJ a system of sets in E for which · (a) Hi, H 2 E SJ implies H 1 u H2 ES); (b) if H 1 c H 2 E S) and H 1 is dosed, then H 1 E S). Let A &B hold iff there are P and Q such that A c P, B c Q, P n Q = 0 and mar P E S), mar Q E S). Show that 8l' is a proximity; 3. Let such that x EE implies O ;:;; f(x) ;;;; I, x EA impHes/(x) = 0, x EB implies/(x/ = 1. Show that & is a proximity. 4. Let Ebe an arbitrary set and, for 0 # A c E, let ,1:)(A) be a filter in E such that each.of (3.1.10) (a)-(e) holds, finally let ,1:)(0) denote the system of all subsets of E. Let A & B hold iff E - BE ,1:)(A). Show that 8J' is a proximity and that ,1:>(A) (A # 0) coincides with the proximity filter of A with respect to& .. 5. Show that if [E, Sf] is a topological space and, for A # 0, ,1:)(A) denotes the neighb.,ourhood filter of the set A, then the statements ·under (3.LlO) are all valid with the exception of (b).

'(3.1.50)

3.1.h.

EXERCISES

135

6. Let [E, pl be a pseudo-metric space and Sl' the proximity defined in exercise 1. Show that ST, = STp, but 1n general SJ' =I= Sl'P. [[E, p] = [R, pi], A is the set of even numbers, B is the set of odd numbers.] 7. tet E = R, p1(x, y) = Ix - y I, p(x, y) = I 0) constitute a uniform subbase on E. Let us assign now to every finite subset 0 ,f. I:' c I: and every e > 0 the set · (3.2.16)

UI',• = {(x, y): a(x, y) < e,

if a EI:'} C Ex E.

(3.2.19)

3.2.d.

UNIFORMITY INDUCED BY A FAMILY OF PSEUDO-METRICS

If I;' runs over all finite non-empty subsets of I; and numbers, denote the system of sets obtained by U2 .

8

141

runs over all positive real

(3.2.17) If I; # 0 is a family of pseudo-metrics on E, then the system (3.2.18) U;; = {Ux,,,: 0 #

I;'

c I: is finite,

8

> O}

of sets (3.2.16) is a uniform base on E. The uniformity "11;; generated by U;; is called the uniformity induced by the family of pseudo-metrics I:. Pro9f. Ac.cording to (3.2.15), the finite intersections of the sets U",' constitute a uniform base. On the other .hand, U;; is . equivalent to this since, for '

n

=

Ux',,

moreover if O
Y, :r be a family of pseudo-metrics on Y and let a* be, for a E :r, the pseudo-metric on X defined by the formula (3.2.42)

a*(x, y)

=

a(f(x),f(y)),

(3.2.46)

3.2.h.

147

INVERSE IMAGE OF UNIFORMITIES

Proof. It is clear that u* in (3.2.42) is a pseudo-metric together with u. If 0 # # I'1 c I' is finite and rt = {u*: u E L'1 }, then evidently

Ur~,•= {(x,y):u*(x,y) < B ifu*EI"r} = = {(x, y): u(f(x),f(y)) < B if u EI'1 } = -- 9 -1(Ur,,e) '

where 9 is the map which occurs in (3.2.39). As the filter base composed of the surroundings Ur,,, generates 611'.r, while the one composed of Uxf,• does the same for 611'.r•, the statement follows from (2.6.9) (b). I A generalization of (3.2.35) is: (3.2.43) If f: X

-+

Y and 6U is a uniformity on Y, then

1-1(&'1J) =

&f-l('IJ),

1-1(fJ'IJ) =

ff/-l('IJ).

Proof. If A, B c X and U E611'. is a surrounding, then (Ax B) n 9- 1(U) # 0 is equivalent to, (f(A) xf(B)) n U # 0. I

As a consequence of (2.6.9) (a): (3.2.44) Jff: X-+ Y, 611'.1 and"U'. 2 are uniformities on Y, 611'. 1 < 611'. 2 , thenJ- 1 (611'. 1 )
, < "Uc:!>, < "Uc:!>, 14. With the notations of the preceding Exercise 13, let E = R, )8 be composed of all subsets of R. Show that (a) &'ll0, = &'ll0, = Si''l!.,, = Si'; (b) Si' is identical with the discrete proximity of R;

3.2.k.

EXERCISES

159

(c) '\£0, =I '\£0, =I '\£0,; (d) '\£0 . is the discrete uniformity . . 15. With the notations of exercise 13, let E = Rand the system mbe composed of the intervals of the form ( - oo, + oo), ( - oo, b), [a,+ oo ); [ c, d), (a, b, c, d ER, c < d). Show that (a) 8i'tl.,, = 8i'll.,, = 8i'll.,, = $+; (b) 8i''ll.,, "# 8i''ll.,,; (c} '\£0, = '\£0,· [If A denotes the set of all even, B that of all odd· numbers, then A 81''11.,, B but Ai'll.,, B.] 16. Let [E, 8i] be a topological space in which the open-closed sets constitute a base, S) = {0}, and 8i' the proximity studied in exercise 3.1.h.9~ Show that if the system mof exercise 13 is coin.posed of all open-closed sets relative to 8i, then

17. Under the hypotheses of exercise 13, let 0 "# Eo c E, m0 = m(n) {E0 }, t and r' - x, then, if RE t and VE tl(x), there exist an R~ E t' such that R{ c V, and R~ E t' such that R; c R, and finally an R~ E t' with the property 0 =f R; c R~ n R~ c V n R. Accordingly x E R for all R E t. (e) => (a): is obvious. I ·

(5.2.30)

5.2.d.

191

CLUSTER POINTS OF FILTER BASES

It follows at once from (5.2.24): (5.2.25)

If t

--+

x, then x is a cluster point of the filter base r.

(5 2.26) If x is a cluster point of the filter base t and · point of t 1 as well. I

t:1

I

< t, then x is a cluster

(5.2.27) Let [E, Sf] be a topological space, 0 ¥= E0 c E, r a filter base in E0, x EE0 • The point x is a (Sf I E 0 )-cluster point oft if! it is a fl-cluster point. Proof. Under our conditions tJ(x) (n) t = tJ(x) (n) {E0 } (n) t. I

The converse of statement (5.2.25) is of importance: (5.2.28) If r is a compressed filter base in the proximity space [E, Sl'] (in particular, if t: is a Cauchy filter base in the uniform space [E, 61,£]) and x is a cluster point of t, then t: --+ x. Proof. If Vis a neighbourhood of x, i.e. if x Si' E - V, then by (P6 ) there are P and Q such thatPn Q = 0, {x} iE - P, E - ViE - Q. Hence Pis a neighbourhood of x as well and intersects therefore every set R E r. On account of P c E - Q @ E - V, there is one among the sets in t which intersects at most one of the sets P and E - V, thus it intersects only P: RE rand R c V, I

The following statement is similar to .the preceding one: (5.2.29) Let [E, fl] be a topological space, u an ultra.filter in E, x a cluster point of u. Then U --+ x. Proof. Suppose that Vis an arbitrary neighbourhood of x. then by (5.2.11) either VE u or E - VE u. The latter contradicts the fact that x is a cluster point

of

u.1

-The following theorem can Iiow be proved: (5.2.30) A proximity space is .compact if! every filter base admits a cluster point. Proof. If [E, Sl'] is compact and tis a filter base in E, then by (5.2.15) there is an ultrafilter u in E containing t (thus finer than t). On account of (5.2.17), u is compressed, hence convergent: u --+ x. According to (5.2.24), xis a cluster point

of r. On the other hand, if every filter base in E has a cluster point and r is a compressed filter base,' then r converges to any of its cluster points according to (5.2.28),

I

Theorem (5.2.30) shows indeed that the compactness of a proximity space depends only on the topology of the space as the existence of. a cluster point of a filter base is determined by the neighbourhood filters of the points. Moreover, in connection with (5.2.30), we have the possibility of defining the compactness of topological spaces in a manner that is in accordance with the compactness of · proximity spaces earlier defined: let us call the topological space [E, fl] and the

5.2.

192

(5.2.31)

COMPACT PROXIMITY SPACES

topology /if compact if every filter base in E has a cluster point. Using this terminology, we can formulate (5.2.30) as follows:

(5.2.31) The proximity space [E, Si'] is compact

if! the topology iifsr is compact.

I

5.2.e. Exercises. 1. Let Si' be the proximity defined in exercise 3.1.h.2. Show that a filter 5 in E is compressed iff mar P E S) implies either P E 5 or E - P E 5. 2. Let where tJ_j denotes the function family consisting of all @-continuous functions. I

The properties of functions continuous on a bounded closed set, well known from elementary analysis, can be generalized in the following way: (5.3.27)/n a compact space every continuous function takes on a greatest and a least value. Proof. If [E, 31] is a compact space and f: E--+ R is continuous, then g = = f /1iE) : E = f(E) is a (31, $ I/(£))-continuous surjection on account of (2.6.21), so that by (5.3.10) f(E) is a compact subset of R, hence it is bounded and closed by (5.3.8); therefore supf(E)and inf/(E) exist and belong tof(E), I (5.3.28) If [X, &] and [Y, 0] are proximity spaces, §' is compact and f: X--+ Y is (@&, 310 )-continuous, then f is(&, Q)-proximally continuous.

(5.3 31)

5.3.c.

199

COUNTABLY COMPACT SPACES

Proof. According to (3.1.15), W@ is (S 3) and a fortiori (S 2), moreover compact on account of (5.2.31). According to (5.3.25), SY is the unique proximity inducing W@, thus it coincides with the Cech-Stone proximity of W@, Hence (3.1.49) can be applied. I

Similarly, as a generalization of a classical theorem of analysis, the following is true: (5.3.29) If [X, 611i] and [Y, 611 2 ] are uniform spaces, 611 1 is compact andf: X-+ Y is (8!"11,, 8!"11)-continuous, then f is (6111, 611 2)-uniformly continuous. Proof. f is (&"11,, &"11 )-proximally continuous according to (5.3.28), thus g = = f 11),XJ: X-+ f(X) is (&"11,, &"11, I f(X))-proximally continuous by (3.1.47), and it is (8f"l1, 8!"11 [f(X))-continuous according to (2.6.21). On account of (5.3.10), 8!"11 lf(X) is ~ompact, and therefore 611 2 lf(X) is precompact by (5.2.31) and (5.2.21). Hence g is (611 1 , 611 2 lf(X))-uniformly continuous by (3.2.77), while f is (6111, 611 2)-uniformly continuous according to (3.2.56). I 2

2

.

5.3.c. Countably compact spaces. We shall be occupied with various generalizations of the notion of a compact space. One of them is the following: a topological space (or a topology) is said to be countably compact if every countable filter base has a cluster point. · (Notice that, in contrast to the terminology used here, countably compact spaces are sometimes called compact while compact spaces are said to be bicompact.) As is evident from the definition:

(5.3.30) Every compact topological space is countably compact.

I

Various equivalent characterizations can be given here too: (5.3.31) Let [E, W] be a topological space. The following statements are equivalent to each other ( and to the fact that W is countably compact): (a) Every countable filter base in E has a cluster point; (b) ror every countable centred system c in E,

[n {C: CE c}

=J 0;

(c) For a monotone decreasing sequence F 1 -::::>F2 -::::>F3 -=:> ... of non-empty closed co

sets in E,

n F; =J 0; 1

(d) From every countable open cover of E there can be selected a finite cover; (e) Every sequence of points in E has a cluster point. n

Proof. (a)=> (b): The sets of the form

n C; (C; Ec) constitute a countable filter 1

base t. If x is a cluster point of t, then

xE

n {R:

REt} c

n {C:

CE c}.

200

5.3.

(5.3.32)

COMPACT TOPOLOGICAL SPACES

(b) => ( c): The system {Fi: i E N} is a countable centred system. 00

(c)

=>

(d): Let E = U G;, Gibe open. The hypotheses of (c) are fulfilled by the 1

n,

sets Fn = E -

LJ1

G;, if there cannot be selected a finite cover from the cover '

00

{Gi: i EN}. Hence

n Fn =I= 0 1

00

or

LJ

Gi =I= E would follow.

1

(d) => (e): If the sequence (xn) did not have a cluster point, then, with the notation R; = {xn: n ~ i}, the sets int (E - Ri) would cover E. These sets constitute a monotone increasing sequence so that if a finite number of them cover E, then one of them covers it, hence int (E - Rn) = E, Rn = 0 for some n: this is a contradiction. n

(e)

=>

(a): Let r = {Ri: i EN} be a countable filter base in E aJ?,d Xn E

n Ri. 1

If x is a cluster point of the sequence (xn), then, for every m E N and every neighbourhood V of x, there is an n ~ m such that Xn E V and then Xn E Rm n V. Thus x is a cluster. point of r. I As a generalization of (5.2.21), the following holds: (5.3.32) Every countably compact uniform space is precompact. Proof. Suppose that [E, 6U] is a uniform space which cannot be covered, for a given surrounding U E 6lf, with a finite number of sets small of order U, and let U1 E6lC be a surrounding such that U1 o Ui c U. Starting from a point x1 EE, n

let Xn+l EE be a point such that Xn+l

~

U Ui(xl

As Ui(xi) is small of order U,

1

an Xn+l of this property can certainly be found according to our hypothesis. The sequence obtained in this manner cannot have any cluster points as n =I= m implies (xn, Xm) ~ Ui, so that, if U2 E 6lC is a smrounding such that U2 o U2 c Ui, then Xn E Ulx) can hold, for any point x EE, at most for one n. Therefore the space cannot be countably compact. I The important fact that, in the case of pseudo-metric spaces, the notion of compactness coincides with the notion of countable compactness can be proved with the help of the preceding theorem: (5.3.33) A pseudo-metric space is compact iff it is countably compact. Proof. On account of (5.3.30), it only remains to be proved that if the pseudometric space [E, p] is countably compact,. then it is compact. It is known from (5.3.32) that the uniformity 61,CP of the space is precompact, we shall show that it is complete and then (5.2.22) implies the assertion. On account of (5.1.10), it is sufficient· to show that [E, p] itself is complete. However, if (xn) is a Cauchy sequence in E and xis a cluster point of (xn), then by (5.2.28) Xn ~ x. I

Similarly there follows at once from (5.3.31) (d): (5.3.34) A Lindelof space is compact ifj' it is countably compact.

I

(5.3.43)

5.3.c.

201

COUNTABLY COMPACT SPACES

Notice that by (5.3,32) and (3.2.68): (5.3.35) Every (countably) compact pseudo-metric space is separable.

I

One part of the properties of compact spaces can be extended to countably compact spaces. In order to do this, a set Kc E will be called countably compact in the topological space [E, 3f], if K = 0 or K ¥= 0 and 3f I K is countably compact. The properties enumerated in (5.3.31) can be formulated for this case too: it will be sufficient to mention the following of them: · (5.3.36) K is countably compact in the topological space [E, 3f] if! every sequence ofpoints in [(admits a 3i-cluster point in K, or iff there can be selected a.finite cover of [(from every countable 3i-open cover of K . Proof. x EK is a 3f-cluster point of the sequence (xn) in J( iff it is a (3f I K)cluster point. On the other, hand,. the (3f I K)-open covers of K coincide with the covers obtained by intersecting K with the members of 3f-open covers. I (5.3.37) Every closed subset of a countably compact set is countably compact as well. Proof. If K is countably compact, F c K, Fis dosed and ·(xn) is a sequence in F, .then (xn) has a cluster point x EK. Evidently x EF = F. I

According to (5.3.6), it can be proved that: (5.3.38) The union of a finite number of countably compact sets is .countably compact as well. I

On the analogy of (S.3.10), it is the case that (5.3.39) A continuous image of a countably compact topological space is countably compact. I (5.3.40) If 8T1 and 3f2 are topologies on E, 3i1 . < 3f2 and 3f2 is countably compact, then 3i1 has the same property. I ·,

Referring to (5.3.39) and (5.3.33) instead of (5.3.10), it can be proved Eimilarly to (5.3.27) that: (5.3.41) In a countably compact space, every continuous function takes on a greatest and a least value. I

A topological space is said to be pseudo-compact if every continuous function is bounded iti it. It follows at once from (5.3.41) that: (5.3.42) Every countably compact space is pseudo-compact,

I

According to (5.3.5): (5.3.43) If [E, 3f] is a T 2-space and an M 1-space, then every countably compact set is closed.

202

5.3.

(5.3.44)

COMPACT TOPOLOGICAL SPACES

Proof. If Khas this property, then by (2.4.10) it is sufficient to show thatxn EK and xn ~ x EE imply x EK. Applying the reasoning of the proof of (5.3.5) for the sequential filter base r belonging to (xn), the statement results easily. I

Applying, in the proof of (5.3.13), (5.3.37) instead of (5.3.4), (5.3.39) instead of (5.3.10), and (5.3.43) instead of (5.3.5), we get: (5.3.44) Let [X, 8T1 ] and {Y, 8T2 ] be two topological spaces,!: X ~ Ya continuous bijection, 8T1 countably compact, 8T2 a T 2-topology and an M 1 -topology. Then f is a homeomorphism. I

5.3.d. Sequentially compact spaces. The property of the bounded closed sets of the space Rm stated in the Bolzano-Weierstrass theorem motivates the following definition: a topological space or a topologf is said to be sequentially compact if every sequence of points has a convergent_ subsequence. It follows at once from (5.2.24) and (5.3.31) that: (5.3.45) Every sequentially compact space is countably compact.

I

By way of converse, only the following can be said: (5.3.46) Every countably compact Mi-space is sequentially compact. Proof. Let (xn) be an arbitrary sequence in the space [E, 8T], and x a duster point of (xn). Using (2.4.13), let {V.,,: i EN} be a neighbourhood base of x such that Vi:::>. V2 ::) V3 ::) ••• , and let. us construct the subsequence (xn,) in the foHowing way: let Xn, E Vi, and if Xn1_, is chosen, let ni > ni-l such that Xn1 E v;. Evidently xn, ~ x. I ·

As a pseudo-metric space is always an Mi-space, by (5.3.45), (5.3.46) and (5.3.33) it can be said: (5.3.47) A pseudo-metric space is compact

if! it

is sequentially compact,

I

The subset K of the topological space [E, 8T] is said to be sequentially compact if K = 0 or if K =f, 0 and 8T 1.Kis sequentially compact. Then similarly to (5.3.37) it can. be proved: (5.3.48) A closed subset of a sequentially compact set is sequentially compact as

well.

I

·

·

(5.3.4.9) The union of a finite number of sequentially compact sets is sequentially compad as well. m

Proof. If Ki is sequentially compact and K =

LJ K;, then a suitable subsequence 1 .

of an arbitrary sequence (xn) in K is contained in a member K; and a subsequence of this converges to a point in K;. I (53.50) A continuous image of a sequentially compact space is sequentially compact as we!!:

(5.3.56)

5.3.d.

203

SEQUENTIALLY COMPACT SPACES

Proof. Let f: X - Y be a continuous surjection. If (Yn) is an arbitrary sequence in Y, let Xn E X be such that f(xn) = Yw If Xn; - x, then by (2.6.13) Yni = f(Xn;) -f(x). I

(5.3.51) Jf 8J1 and 8T2 are topologies on E, 8T1 < 8T2, and 8T2 is sequentially compact, then 8T1 has the same property. I '

.

\

5.3.e. Locally compact spaces. It is a very strong restriction for a space td be compact; e,g. this restriction is not fulfilled by the space [Rin, ~m]. Nevertheless, it is true .that every point has a compact neighbourhood in Rm: the closed sphere S(x, e) is, for any point x E Rm and any e > 0, a compact neighbourhood of x on account of (5.3.3). Accordingly, it is worth-while studying the topological spaces [E, 8T] in which every point has a compact neighbourhood, a space having. this property (or its topology) is said to be locally compact. According to the previous remark: (5.3.52)The space [Rm, ~m] is locally compact,

I

A further example of a locally compact space is any discrete space. (5.3.53) A closed subset of a locally compact space is locally compact as well. Proof. If, in the space [E, 8TJ, 0 =I= E0 cEjs closed and Kis a compact Sf-neigh.bourhood of x EE0 , then Kn E0 is (Sf I K)-closed and hence by (5.3.4) (8T I K)compact, thus ((Sf I K) I Kn E0)-compact, thus (Sf I Kn E0) compact and therefore ((Sf I E0J I KnE0)-compact; here we applied (2.3.17) repeatedly. Thus KnE0 is a compact neighbourhood of x in the subspace [E0 , Sf I E0 ]. I 0

Locally compact S2-spaces have extremely useful properties. (5.3.54)1n a locally compact S 2-space, every point has a neighbourhoodbase con· sisting of compact closed sets. Proof. Let K be a compact neighbourhood of x in the S 2-space [E, Sf]. By (5.3.20), it can immediately be supposed that K is closed. Let V be an arbitrary neighbourhood of x and G = int Vi Then 1( - G = (E - G) n K is compact and closed by {5.3.4), hence by (5.3.17) there exist open sets Viand G1 such that x E Vi, K - G c G1 , Vi n G1 = 0. Then, by 't\ c E ~· G1, V1 n K is a compact closed neighbourhood of x such that V1 n Kc G c V, I

(5.3.55) An open subspace of a locally compact S 2-spac~ is again locally compact. Proof. If in. the locally compact S2-space [E, Sf], x EE0 c E, and E0 is Sf-open, then by (5.3.54) there exists a Sf-compact K such that Kc E0 , and which is a 8f-neighbourhood of x. This K is of course a (Sf I E0)-neighbourhood of x and is (Sf I E0)-compact too. I Conversely it is the case that: (5.3.56) In a T 2-space, every locally compact and dense subspace is open.

204

5.3.

(5.3.57)

COMPACT TOPOLOGICAL SPACES

Proof. Let [E, 8!] be a T2 -space, E0 c E dense, and 8i [ E 0 locally compact. If

x E E0 , then there exists a compact (§" I E0)-neighbourhood K of x. Let G be a . 8!-open set such that x E G n E 0 c K. Since E0 is dense, for each 8!-open neighbourhood V of any yE G, we have V n GnE0 ,f. 0, yE GnE0, hence G c GnE0 c K as K is 8!-closed by (5.3.5). Therefore x E G c Kc E0 , x E int E0 • I It can be seen from (5.3.54) that every locally compact S2-space is regular. What is more, these spaces are completely regular. This will be shown by giving a proximity inducing the topology of the space: (5.3.57) Let [E, 8!] be a locally compact S 2-space and let A & B ijJ An jj = 0 and at least one of the sets A and jj is compact. Then & is a proximity inducing. the topology 8!; more precisely, it is the coarsest of the proximities inducing 8!. Proof. (P1) and (P2) are evidently fulfilled by the relation&. If A' & B', Ac A' ' B c B', then A & B holds; for, if e.g. A' is compact, then, by A c A' and (5.3.4), A is compact as well; thus (P3) holds. (P4) follows from the fact that 0 = 0 is compact. A sf C and B sf C imply A n C = jj n C = 0, thus A u B = A u jj implies A u B n C = 0, and if C is compact, then A u B & C evidently holds while, if C is not compact, then A and B, and by (5.3.6) Au jj = Au B are compact. Accordingly (P5) is fulfilled. Finally assume An jj = 0, and, say, let A be compact. As a consequence of (5.3.54), there can be given; for every point x EA, a .

. compact closed neighbourhood Kx such that Kx n B = 0. The compact set

A

n

is covered by a finite number of sets int Kx: n

Let K

=

Ac U int Kx;,

X;

EA (i = I, .. , n).

l

U Kx;· Then by (5.3.6) K is compact, closed, Kn jj

=

0, and

Ac

int K.

1

With the notations P = int K, Q = E - K, we have P n Q = 0 and An E - P = = 0, B n E - Q = 0, finally A and E - Q = K = K are compact. Thus (P6 ) is fulfilled. On account of the foregoing, it can be said that & is a proximity on E. If x EE and Vis a proximal neighbourhood of {x }, then x n E - V = 0, thus x ~ E - V . and E - E - V c Vis a neighbourhood of x. Conversely, if Wis a neighbourhood of x, then, by x Eint W, x ~ E - int W, and, on account of (2.5.8), x n E - int W= = 0, so that xis compact by (5.3.15). Therefore {x} sf E - int W, and a fortiori {x} &E - W, Wis a proximal neighbourhood of {x}. Hence & induces the topology §". Finally let &' be an arbitrary proximity inducing 8!. If A & B, then by (5.324) A &' Band a fortiori A &' B. Thus & < &'. I 5.3.f. Rim-compact spaces. We shall consider spaces having properties similar, from many points of view, to those of locally compact spaces, but more general at least in the case of S2-spaces.

(5.3.61)

5.3.f.

205

RIM-COMPACT SPACES

The topological space [E, 81] (or the topology 3f) is said to be rim-compact if there is a base consisting of sets with compact boundary. (5.3.58) Every locally compact S 2-space is rim-compact. Proof. If the space [E, 81] has this property and Vis an arbitrary neighbourhood -of x EE, then there exists by (5.3.54) a compact closed K s.uch that x E int Kc V. With the notation G = int K, on account of (2.2.20), mar G c G c K and mar G is closed so that it is compact by (5.3.4). Thus the open sets with compact boundary constitute a base for 81. I

© I Q is a further example for a rim-compact topology; the sets of the form J n Q, where Jc R is an open interval, constitute a base for it and the boundary of each of these sets consists of two points at most, thus it is compact. It can be seen from (5.3.56) that ©I Q is not locally compact as Q is @-dense without being @-open. Observe that those sets In Q already constitute a base for $ I Q for which I is an open interval with irrational end-points; their boundary with respect to ©I Q being empty, they are open-closed sets by (2.2.22). A topological space or a topology having a base consisting of open-closed sets is called zero-dimensional. As a generalization of the preceding example, it can be asserted at once: (5.3.59) Every zero-dimensional space is rim-compact.

I

Similarly to (5.3.53), it is the case that: (5.3.60) A closed subspace of a rim-compact space is rim-compact as well. Proof. IfE0 #-1 0 is closed in the topologicalspace [E, 81], G is open and mar G is compact, then, since G n E 0 is (81 I E0)-open, the (81 I E 0)-boundary of this set is H = (G n E 0 n E 0) - (G n E 0) c G - G = mar G on account of (2.2.20) and since H is (81 I E0)-closed, hence 81-closed, it is compact on account of (5.3.4).1 Instead of (5.3.54), the following can be said: (5.3.61) Every rim-compact S 2 -space is regular. Proof. If Vis a neighbourhood of x, let x E G c V, where G is open, and mar G is compact; x and any point of mar G are weakly separated and therefore disconnected. To every pointy Emar G, let us assign an open set Vy such that x ~ Vr Then n

mar G c

U

Vy; (Yi, Emar G, i

= l, ... , n).

1

Let 1V

W = Gn

n (E -

Vy)·

1

This is an open neighbourhood of x and W c V, as W c G, the points of G belong to V, while every point of mar G lies in a Vy;, and thus cannot belong to W since w CE - Vy; C E - Vy;, w C E - VYI' I

5.3.

206

(5.3.62)

COMPACT TOPOLOGICAL SPACES

Similarly to (5.3.55): (5.3.62) An open subspace of a rim-compact S 2-space is rim-compact .as well. Proof. If E0 =fa 0 is open and, for a given Sf-open neighbourhood V c E 0 of x E £ 01 G is a Sf-open set with a compact&f-boundary such that x E G c G c V (by (5.3.61) there exists a set with this property), then G is (Sf £ 0)-open and its (SJ l E0)-boundary is by (2.2.20) , J

(GnE0) thus it is compact.

-

G

= G-

G

= mar G,

I

The following remarks will be useful later:

B,

(5.3.63) If A and Bare sets with compact boundaries, then each of the sets Au A n B, A - B has a compact boundary as well. · Proof. By (2.2.20), the boundary of each of the sets in question is closed and, by (22.21), it is a subspace of mar A u mar B, thus it is compact by (5.3.6) and (5.3.4), I . (5.3.64) If A is a set with compact boundary, then int A and A are also of compact boundary. Proof. On account of (2.2.20), ·mar i:nt A and mar A are closed, mar. int A == = int A - int A. c A -:- int A = mar A, mar A = A - int A c A - int A = = mar A, I

The purpose of the following rea,soning is to show that every rim-compact S2-topology can be induced by a proximity and is therefore completely regular; (5.3.65) Let [E, Sf] be a rim-compact S 2-space and denote by \:f3 the system of . open sets with compact boundary, by Q the system of closed sets with pompact boundary. Then . (a) PE l;J3 implies E - PED,; (b) Q E O implies£ - Q E l;J3; (c) P 1, P 2 E $ implies P 1 u P 2 E $, P 1 n P 2 E $; (d) Qi, Q 2 ED, implies Q1 u Q 2 E 0, Q1 n Q2 E O; (e} if Kis compact and closed, x EE - K, then there are Pi, P 2 E $ such that x EPi, Kc P 2, l\n P2 = 0; . (f) if K1 and K2 are compact, one of them closed, and K1 n K2 = 0, then the,;e .are P 1 , P 2 E $ such that K 1 C P 1 , K 2 C P 2, P1 n P2 = 0; (g) if x E PE.$, then there is a Q E Osuch that x E Q c P; (h) if Qi, Q2 E0, Q1 n Q 2 = 0, then there are Pi,P2 E $ such that Q1 c Pi, Q2 c P2, P1 n P 2 = 0. , Proof. (a)-(d) result from (5.3,63), since mar E = 0 implies EE$ n 0. (e): x and everyyE Kare weakly separated, hence disconnected; Since $ is a base and the space is regular by (5.3.61), we can find, for y EK, sets Pi(y), .

'

.

'

Ply) E $ such that x E Pi(y), YE Ply), Pi(y) nPly)

i

= 0. Let

h

'

Kc LJP2(y1). 1.

(5.3.66)

5.3.f.

207

RIM-COMPACT SPACES

Then n

P1

=

n Pi(Y;), P

n

2

= LJ Ply;)

1

will do; for by (c) Pi, P 2 E 1,13 and

J\ c

Pi(y;)

1

c E - Ply;),

l\

n

=

LJ

P 2(y;) imply

1

P1 n P2 = 0. (f); E.g. let K2 be closed. Then by (e) there are, for x

EK1 , sets Pi(x), P 2(x) E 1,13 n

such that xEPi(x), K 2 c Plx), P 1(x) n Plx) = 0. If K 1 c

U Pi(x;),

then

1

n

P1 =

LJ

n

Pi(x;), P 2 =

n P (x;) 2

1

1

will do. (g): On account of (e), let P 1, P 2 E 1,13 be such that x EPi, mar Pc P 2 , P1 n P2 = = 0. Then Q = P - P 2 = P - P 2 = P n (E - P 2) will do since by (5.3.64) PE :D thus Q E:D by (a) and (d). (h): Since mar Q 1 n mar Q2 c Q 1 n Q2 = 0, therefore by (f) there are P{, P~ E 1,13 such that mar Q 1 c P{, mar Q 2 c P~, P{ n P~ = 0. Let P 1 = (Q1 u P{) n (E - Q 2), P2

= (Q 2 uP~) n (E - Q 1).

Then Q1 u P{ = int Q1 u P{ E 1,13, since by (5.3.64) int Q1 E 1,13, so that, on account of (b) and (c), P 1 E 1,13. In the same way P2 E 1,13. Clearly Q 1 c P 1, Q 2 c P2 , and finally P 1 n P 2 = (Q1 u P{) n (E - Q 2) n (Q 2 u P~) n (E - Q1) =

= P{ n (E - Q2) n P~ n (E - Q1)

C

P{ n P~

= 0.

I

Let us remark that a system of sets 1,13 satisfying the above condition (c) is said to be a lattice. Our purpose will be achieved ~ith the help of the following theorem: (5.3.66) Let 1,1s and :D be two systems of sets consisting of subsets of E # 0 for which 0 E 1,13 n :D, and fulfilling the conditions (a), (b), (c), (d), and (h) in (5.3.65). Define the relation SY in such a way that A §' B holds if! there exist Qi, Q 2 E:D with A c Qi, B c Q2 , Q 1 n Q2 = 0. Then SY is a proximity on E. If moreover gr is a topology on E for which 1,1s is a base and (5.3.65) (g) is fulfilled, then gr = gr&· Proof. (P1), (P 2), (P 3) are evident. (P4) follows from the fact that 0 E:D and by (a) EE :D. (P5 ) can easily be obtained from (d). (P6 ) is true because A SY B implies

A. c Qi,

B c Q 2,

Q1 n Q 2 = 0,

Q1, Q2 E:D,

5.3.

208

COMPACT TOPOLOGICAL SPACES

(5.3.67)

thus by (h) Q1 C P1 E l,js,

Q2 c P2 E l,js,

P1 n P2 = 0,

and on account of (a) A c Qi, E - P 1 E0, Q1 n (E - P 1) = 0, B c Q2 , E - P 2 E 0, Q2 n (E - P 2) = 0, so that A @ E - Pi, B & E - P 2• Therefore & is in fact a proximity on E. If our complementary conditions are fulfilled, and Vis a $-neighbourhood of x, then there is a P E l,j5 such thatx E P c V and, on account of (g), a Q E O such that x E Q c P, and then by

{x}cQ, E- VcE-PEO {x} Sf E - V, so that Vis a 8lg-neighbourhood of x, Conversely, if {x} SJ' E - V, then x E Q1 c E - Q 2 c V, where Qi, Q 2 E 0, thus, together with the $-open E - Q 2 E l,js, Vis a $-neighbourhood of x. I On account of (5.3.65) and (5.3.66), it can now be asserted: (5.3.67) Let [E, 8!] be a rim-compact S 2-space, and let A & B hold if! there are closed sets Q1 and Q 2 with compact boundaries such that A c Qi, B c Q 2, Q1 n Q 2 = = 0. Then & is a proximity on E and 8i = 8i@· I This & is said to be the Freudenthal proximity of the space. 5.3.g. Exercises. I. Let Ebe an ordered set with more than one element, 8f the order topology of E (see exercise 2.2.g. 8). Show that 8J is compact iff there are smallest and largest elements in E and the order of E is complete (see exercise 2.3.c.14). [We have to consider the open cover consisting of the sets ( ~, x) (x EE) if there is no largest element in E, that of the sets (x, -+ ), if there is no smallest element, that of the sets ( ~, x) (x E A) and (y, -+) (y is an upper bound of A) if A c c Eis bounded above but has no smallest upper bound. If, on the other hand, the condition is fulfilled, consider an open cover of E and those x for which, denoting the smallest element by a, [a, x] can be covered with a finite number of the given sets, then the least upper bound of these x.] 2. Let [E, 8i] be a compact topological space,

s(A) c s(B),

s(A n B) = s(A) n s(B)

(A, B c E),

and of course s(E) = E'. Hence it can be seen that the conditions of (2.2.8) are fulfilled by® so that ®is in fact a base for a topology &f' onE'.

(6.1.8)

6.1.b,

STRICT EXTENSIONS

215

The Si'-neighbourhood filter IJ'(x) of the point x EE' is generated, by (2.2.7), by the system of sets {s(G): G is Si-open, x E s(G) }, · and since x E s(A) is equivalent to A E Mx) by (6.1.3), this can be also written in the form · {s(G): G is Si-open, G H(x)}. The trace filter IJ'(x) (n) {E} will be generated on account of (2.1.17) by the system {s(G) n E: G is Si-open, GE ~(x)}.

(6.1.6)

If x EE, then 5(x) (6.1.7)

=

IJ(x), so that G E &(x) holds for a Si-open set G iff x E G, i.e.

s(G) n E = G. (G is Si-open).

Therefore the system (6.1.6) is nothing other than the system of Si-open sets in &(x), which generates &(x), since Mx) is a Si-open filter. Therefore IJ'(x) (n) {E} == &(x),

in particular, if x EE, then IJ'(x) (n) {E} = IJ(x). This shows that .Si' is indeed an extension of Si furnishing the given trace filters 5(x). Now let Si~ be another topology on E', IJ~(x) the Si~-neighbourhood filter of the point x EE', and suppose that, for each point x E E', IJ~(x) (n) {E}

= &(x).

If x E s(G) and G is Si-open, then, by GE 5(x), x has a sr;.neighbourhood v; such that G = v; n E and theri there exists a Si~-open set G; such that x. E G; c v;. If y E o;, then G; E o;(y), thus G; n EE 5(y), and G; n E c v; n E = G implies GE 5(y), y E s(G). Therefore if G is Si-open, s(G) contains a sr;-neighbourhood o; of any point x E s(G) so that s(G) is sr;-open. This shows that every &f'-open set is m;-open as well, @' < sr;.1 6.1.b. Strict extensions. The extensions arising in the way described in theorem (6.1.2) are called strict extensions. More precisely, the topological space [E', Si'] is called a strict extension of the topological space [E, Si] (or the topology Si' a strict extension of the topology Si) if E c·E', Si= Si' IE', Eis Si'-dense, and if, denoting by &(x) for x EE' the trace filter IJ'(x) (n) {E} of the Si'-neighbourhood filter IJ'(x), and for A c E by s(A), the set in (6.1.3), the system of sets ® in

(6.L4) is a base for Si'. With this notation we can see from (6.1.2): (6.1.8) Let [E, Si] be a topologicalspace, E' ~ E, and let us assign to every point x EE' a Si-open.filter &(x)inE and suppose that 5(x) is the Si-neighbourhood.filter of x for .x EE. Then there, is a unique topology Si' on E' which is a strict extension of Si· andfurnishes the given.filters &(x) as trace.filters; this is the coarsest of all topologies on E' leading to the given trace.filters, I

216

6.1.

EXTENSIONS OF TOPOLOGICAL SPACES

(6.1.9)

In order to give a further characterization of strict extensions, let us notice the following: (6.1.9) Let [E', $'] be a topological space, E c E' $'-dense, $ = $'IE, the $'-neighbourhood filter of x EE' be tJ'(x), 5(x) = tJ'(x)(n) {E}, and s(A)

= {x:xEE', AE5(x)}

for A c E. Then (a) if B c E, the $'-closure of the set Bis E' - s(E - B); (b) if G c Eis $-open, then s(G) is the largest $'-open set whose interse(:tion with EisG. Proof. (a): x EE' does not belong to the $'-closure of B iff it is not a $'-limit point of B, i.e. iff there exists in tJ'(x) a set not intersecting B which holds iff there exists in 5(x) a set not intersecting B, i.e. iff E - B E 5(x), x Es(E - B). (b): According to the foregoing E' - s(G) is the $'-closure of the setE - G, i.e. the smallest $'-closed set whose intersection with Eis E - G. Passing to the complements, we obtain the assertion. I

Let us now introduce the following term: in. the topological space [E, $] (or for the topology $), the system %is a closed base if the system {E ,- F: FE %} is a base for $, i.e. by (2.2.6) if the sets FE ij are $-closed and every $-closed set distinct from E can be constructed as an intersection of sets belonging to %. Now we can easily prove: (6.1.10) Let the topological space [E', $']bean extension of [E, $]. $' is a strict extension of$ if.! the $'-closures of the ($-closed) sets B c E constitute a closed base/or$'. Proof. If$' is a strict extension of$; then the sets of the form s(G), where G is $-open, constitute a base for$'; by (6.1.9) E' -s(G) is identical with the $'-closure of E - G so that the $'-closures of $-closed sets constitute a closed base for $'. Conversely, if the $'-closures of sets in E constitute a closed base for $', then the same is true even for the $'-closures of $-closed sets as the $'-closure of a set B c Eis of course identical with the 8f'-closure of the $-closure of B. Therefore, in this case, by (6.1.9), the sets of the form s(G), where G is $-open; constitute a $'-base, so that$' is indeed a strict extension of$, I · · (6.1.11) If, with' the hypotheses and notations of (6.1.9), the topology$' is regular, then it is a strict extension of Sf. Proof. For an arbitrary point x EE' and its Sf'~neighbourhood V' we can find a Sf'-open G' and a $'-closed F' such that x E G' c F'c V'. Let F = (E' - G') n E. Then, by F c E' - G', x cannot belong to the Sf'-closure ofF. On the other hand, if y EE' - V', then by y EE' - F', together with every Sf'-neighbourhood v; of y, (E' - F') n v; is also a Sf'-neighbourhood of y which intersects E since Eis $'-dense, i.e. v; intersects the set (E' - F') n E c (E' - G') n E = F. Hence the complement of the Sf'-closure of Fis a Sf'-neighbourhood of x contained in V'. 'therefore (6.1.10) can be applied. I

(6.1.14)

6.1.b.

STRICT EXTENSIONS

217

(6.1.12) Let EC E' CE", gr, gr,, gr" topologies on E, E' and E" respectively. If gr, is an extension of gr and gr" one of gr', then gr" is an extension of gr as well. Proof. If gr= gr, IE, gr,= gr" IE', then by (2.3.17) gr= gr" IE. If Eis gr,_dense and E' is gr" -dense, then, for every gr" -open neighbourhood G" of any point x EE", G" n E' f:: 0, thus G" n E' being gr, -open, G" n E' n E = G" n E f:: 0, and E is gr" -dense. I (6.1.13) Let EC E' CE", gr" a topology on E", gr' = gr" IE', gr = gr" IE= = gr' IE. If gr" is a (strict) extension of gr, then gr, is a (strict) extension of gr and gr" one of gr,_ Proof. If E is gr"-dense, then its gr, -closure is equal to E" n E' = E' and it is therefore gr' -dense too, further the gr" -closure of E' is E" as well, thus E' is also gr"_dense. If gr" is a strict extension of gr, then the gr" -closures of the subsets of E constitute a closed base for gr" by (6.1.10). Hence the gr"_closures of the subsets of E' consti- · tute a fortiori a closed base for gr" and gr" is a strict extension of gr,_ On the other hand, the gr'-closme of Ac Eis the intersection of E' with the gr"_closure of A; by (2.3.13) (e) these intersections constitute a closed base for gr, so that gr, is a strict extension of gr_ I Let us notice in connection with separation properties of strict extensions: (6.1.14) With the hypotheses and notations of (6.1.9), let gr, be a strict extension of gr and x, y EE'. (a) x and y are weakly separated if.! &(x) f:: &(y); (b) x and y are separated if.f neither of &(x) and &(Y) contains the other; (c) x and y are disconnected if.! 0 E s(x) (n) &(y). Proof. (a): By (2.5.4) x and y are weakly separated iff b'(x) f:: ti'(y). If s(x) f:: f:: s(y), then of course b'(x) f:: ti'(y) holds as well. On the other hand, if b'(x) f:: f:: b'(y), then e.g. x has a gr,_neighbourhood which is not a gr,_neighbourhood of y and there exists a set of the form s(G) where G is gr_open such that x E s(G), y ~ s(G), i.e. GE s(x), G ~ &(y). (b): If neither of &(x) and &(y) contains the other, then, as these filters are open, there exist gr_open G1 and G2 such that G1 E &(x), G1 ~ &(y), G2 E &(y), G2 ~ ~ 5(x). Then s(G1) is a gr,_neighbourhood of x not containing y, s(G2) a gr'-neighbourhood of y not containing x. On the other hand, if x has a gr,_neighbourhood not containing y and y has one not containing x, then these can be taken in the form s(G1) and s(G2) where G1 and G2 are gr-open, and G1 E &(x), G1 ~ 5(y), G2 E 5(y), G2 ~ &(x). . (c): If x and y have disjoint gr,_neighbourhoods V{ and V~, then V{ n EE 0(x) and V~ n EE &(y) are disjoint as well. On the other hand, if 0 E &(x) (n) s(y), then as they are open, there exist gr_open G1 and G2 such that G1 E s(x), G2 E &(y), G1 n G2 = 0; s(G1) and s(G2} will then be disjoint gr,_neighbourhoods of x and y respectively, for by (6.l.5), s(G1) n s(G2) = s(G1 n G2) = s(0) = 0.1 In this connection let us introduce the following term: the.extension [E', Sf'] of the topological space [E, gr] (or the extension gr' of the topology gr) is called a

6.1.

218.

EXTENSIONS OF TOPOLOGICAL SPACES

(6.1.15)

reduced extension if x EE' - E, y EE', x =/. y imply that x and y are weakly separated. It follows at once from the definition: (6.I.15) If the extension [E', §'] of the space [E, SJ] is a T0-space, then it is a reduced ex.tension. I (6.l.16) If [E, §J ts a T 0sspace and [E', §'] is a reduced extension of [E, SJ], then [E', §'] is a T0-space as well. Proof.. Itneed only be shown that if x, y EE; x :;6 y, then x and y are weakly separated with respect to SJ'. But in this case x and y are weakly separated with respect to SJ, thus there exists e.g. a §~open G such that x E G, y EG: For a suitable SJ'-open G', G G' n E and then x E G', y fG'. I

=

(6. l.17) Let [E', §'] be a strict extension of [E, §],for x EE' let 5(x) be the trace in E of the §' -neighbourhood filter of the point x. [E', §'] is a reduced extension of [E, SJ] if! x EE' - E, y EE', x =/. y implies &(x) =/. 5(y). Proof. (6.l.14) (a). I ·

c E' c E'.', §" be a topology on E", §' = §" IE', § = §' IE = E, E §" -dense. If§" is a reduced extension of§, then §' has the same property. Conversely if §' is a reduced extension of §, while §" is one of§', .then §" is a reduced extension of§ as well. Proof. If§" is a reduced extension of SJ and x EE' - E, y EE', x =/. y, then there is a §"~open G" such that e.g. XE G", y EG", thus G" n E' is a sJ'',neighbourhood of x not containing y. · Suppose now that §' is a reduced extension of SJ, §" one of§', and let x EE" -' - E, y EE", x =/. y. If x EE" - E'; then x and y are weakly·SJ"-separated.·If x E EE' ~ E, y EE" - E', the same can again be asserted because of the fact that SJ" is reduced with respect to SJ'. Finally if x EE' ~ E, y EE', then there is a SJ'open G' such that e.g. x E G', y EG' and choosing a SJ"-open G" such that G' = = G" n E', at the same time x E G", y EG" will hold. I (6.1.18) Let E

= §"

J

The content of the following theorem is that strict extensions are essentially defined uniquely by prescribing the trace filters. In order to formulate this more precisely, let [E{, §{] and [E~, SJ~] be two extensions of the space [E, SJ],f: E{ --+E~ a map such that if x EE then f(x) = x; a map of this kind will be called a map fixingE. . (6.1.19) Let [E{, §{] and [E~, §~j be two strict extensions of the topological space [E, SJ], 5i(x) = b{(x) (n) {E}, &lY) = o;(y) (n){E} for x EE{ and y EE~, where b{(x) and b~(y) denote the SJ{- and8J~-neighbourhoodfilter respectively, and letf: E{ --+ E~ be an injectionfixingE.JJfifD =his a (SJ{,§~ lf(E{))-homeomorphism iffy= J(x) implies 5i(x) = &ly). If§~ is a reduced extension of§. and Ji: E{ --+ E~ as well as ( 2 : · E{ --+ E~ are maps fixingB such that / 1 f;tD is a (SJ{, §~ Ifi(ED)-homeomorphism, while / 2 .I 1~ED is a (SJ{, 8'~ IflE{))-homeomorphism, then Ji == A J

(6.1.20)

6.1.C.

THE ALEXANDROFF COMPACTIFICATION

219

Proof. If h is a (W{, 8!~ If(E{))-homoemorphism, then y = f(x) = h(x), x EE{ implies f(b{(x)) = b~(y) (n) {f(E{)}, thus &i(x) = b{(x) (n) {E} = f(b{(x)) (n) {E} = = b;(y) (n) {f(E{)}n {E} = b~(y) (n) {E} = &ly).

Suppose now &i(x) = &lY) whenever x EE{, y = f(x). With the usual notations si(A) = {x: ~ EE{, s2(A)

A E &i(x)},

= {y: yEE;, AEzly)}

the sets si(G) constitute a Sf{-base, the sets slG) a Sf~-base, and the sets s 2 (G) n nf(E{) a §~ If(E{)-base; G runs always over all Sf-open sets. But, by hypothesis, x E si(G), i.e. GE &i(x) holds iff GE &lf(x)), i.e. f(x) E slG) nf(E{) and therefore h is a (Sf{, Sf; If(E{))-homeomorphism. If f 1 and f 2 are homeomorphisms corresponding to the hypotheses, then by the first statement x EE{ implies &ifi(x)) = &2(flx)) = &iCx). But if x EE, then fi(x) = fix) = x, and if x EE' - E, then since Sf~ is reduced, &lfi(x)) = fr 2(Mx)) can hold only iffi(x) =flx) by (6.1.17).1

It was mentioned ,that the study of extensions of topological spaces is particularly important for the construction of compact extensions. For this purpose strict extensions are very suitable since, if a compact extension of a space is known, then the strict extension belonging to the same trace filters is compact as well on account of (6.1.8) and (5.3.11). In this connection let us notice: (6.1.20) With the hypotheses and notations of(6.l.9) let§' be a strict extension of Sf. Sf' is compact if! from any system of Sf-open sets {G;: i EI} such that to every x EE' there is a G; E &(x), a finite subsystem {G;i: j = I, ... , n} having the same property can be selected. Proof. By (5.3.2) (f) §' is comp2.ct iff there can be selected a finite cover from every cover of E' consisting of sets of the form s(G) (G is Sf-open). I Every compact extension [E', 81'] of a topological space [E, Sf] is said to be a compactification of the space (and Sf' one of the topology Sf).

6.1.c. The Alexandroff compactification. As the first application of strict extensions, let [E, Sf] be an arbitrary non-compact space. Consider the ·complements of compact closed sets in E. These are non-empty as supposed and constitute a filter base consisting of open sets by (5.3.6). Denote by ttJ the (open) filter generated by this filter base, and construct that strict extension [E', Sf'] of [E, 81] in which E' arises by adding a single new point w to E, and &(w) = tu. By assigning as &(x) the Sf-neighbourhood filter b(x) to the point x EE, the extension obtained will be compact. In fact, if {G;: i E I} is a system of Sf-open sets for which to every x E E' there is an i E J such that G; E &(x), then this holds for x = w, i.e. there are an i0 E Janda compact 8f-closed Kc E such that E - Kc G;,; to each point x EK there belongs an ix E J such that G;x E b(x), i.e. x E G;x· Let us select a finite cover

220

6.1.

EXTENSIONS OF TOPOLOGICAL SPACES

(6.1.21)

from the cover obtained in this way for the compact set K: n

KC

U

G;x; (xi EK, ix1 E/).

i=l

For the system {G; G;x,, ... , G;x.} it is again the case that, if x EE', one of its members belongs to &(x), viz. G; if x = w or x EE - K, or a G;x; if x EK; thus by (6.1.20) the following statement is proved: 0,

0

(6.1.21) If [E, W] is a non-compact topological space, E' = Eu {ro}, &(ro) is the filter ttJ in E generated by the complements of compact closed sets, then the strict extension [E', .§"'] corresponding to this choice is compact. I

The extension described in (6.1.21) is called the Alexandroff compactification of the space [E, W] (and its topology that of the topology W). It can be seen from (6.1.19) that the Alexandroff compactification of a space [E, .§"J is uniquely determined up to a homeomorphism fixing E. (6.l.22) With the hypotheses and notations of (6.1.21), &(x) c &(w) cannot hold for any point x EE. Proof. Assuming &(x) c &(w), i.e. b(x) c ttJ, let E = U G;, G; W-open. Then, iEl

for an index i0 E /, x E G;, and there exists a compact closed K such that E - Kc G;, The set K is covered by a finite number of G; although E is not compact: a contradiction. I It follows from this on account of (6.1.17) and (6.1.16): (6.1.23) The Alexandro.ff compactification of any space is a reduced extension. If the space is a T 0 -space, then its Alexandro.ff compactification has the same property. I (6.1.24) The Alexandro.ff compactification of an Si-space is again an S 1-space· Proof. We have to prove by (6.1.14) that x, y EE', &(x) c &(y) implies &(x) = = 0(y). This is true if x, y EE as [E, W] is an Si-space. It is known from (6.1.22) that &(x) c &(w) is impossible whenever x EE and then &(w) c &(x) cannot hold either, as by (5.3.15) E - x E ttJ, but this set is of course not a neighbourhood

of x.

g

(6.1.25) The Alexandro.ff compactification of the space [E, 8!] is an S 2-space if! [E, W] is a locally compact Sz-space. Proof. If [E, W] is a locally compact S 2-space, then, again with the notations of (6.1.21), &(x) -:f. &(y) implies 0 E &(x) (n) &(y), which is true whenever x,y EE since 8J fulfils (SJ, and for x EE and y = w as a consequence of the fact that by (5.3.54) x has a compact closed neighbourhood Kand then (E - K) n K = 0, E - KE &(ro), KE &(x). Hence by (6.1.14) [E', .§"'] is an S 2-space. On the other hand if [E', .§"'] is an S2-space, then by (2.5.20) [E, 8!] is an S 2space as well. Now x EE implies by (6.1.22) &(x) -:f. &(w) thus 0 E &(x) (n) &(w) so

(6.1.27)

6.1.d.

221

,\)-FILTERS

that x has a neighbourhood in [E, 8!] which does not intersect the complement of a compact closed set and has therefore a compact neighbourhood as well. I By (5.3.22) we again find from (6.1.25) that a locally compact S 2-space is completely regular (we deduced it previously from (5.3.57)). 6.1.d. SJ-filters. For the sake of further applications of strict extensions let us introduce some new terms and examine their connections. A system of sets is said to be a semi-lattice if Hi, H 2 E SJ implies H 1 n H 2 E SJ. Every filter in Eis a semi-lattice. The open sets or the closed sets constitute semilattices in a topological space. Now let E f:. 0 be an arbitrary set, SJ a semi-lattice in E. The filter 9 in E is said to be an SJ-filter if it has a base consisting of sets belonging to SJ. If [E, 8!] is a topological space and SJ denotes the semi.-lattice consisting of 8f-open sets, then the SJ-filters are identical with the open filters previously introduced; if SJ is the semi-lattice of closed sets, the SJ-filters are called closed filters. If SJ is a semi-lattice in E, the filter 9 in Eis said to be an ultra-SJ-filter in E whenever it is a maximal SJ-filter, that is an SJ-filter such that if 9' is an SJ-filter in E and 9' ::i 9 then 9' = 9. If [E, 8!] is a topological space and SJ denotes the semi-lattice of open or closed sets, we shall speak about ultraopen or ultraclosed filters in E. The notion of an ultra-SJ-filter is a generalization of the notion of ultrafilter: if SJ consists of all subsets of E, then the SJ-filters in E coincide with the filters in E and the ultra-SJ-filters are identical with the ultrafilters. According to this, the basic properties of ultrafilters can be extended to ultra-SJ-filters as well. (6.1.26) Let SJ be a semi-lattice in E, 9 a SJ-jilter in E. 9 is an ultra-SJ-jilter in E iff HE SJ implies either HE 9 or E - HE 9. Proof. If this condition is fulfilled by 9 and 9' ::i 9 is a SJ-filter in E, then, for A E 9', there is an HE SJ such that H c A, HE t)'. Then E - HE 9 c 9' is impossible so that HE 9, and a fortiori A E 9. Thus in this case 9 is an ultraSJ-filter. Conversely, if 9 is an ultra-SJ-filter, HE SJ and E - H ~ 9, then every set from 9 intersects H. The intersections of sets from SJ belonging to 9 with the set H evidently constitute a filter base consisting of sets from SJ and generating an SJfilter 9' in E. tJ' ::> 9 implies f)' = 9, so that H = H n EE 9. I (6.1.27) If SJ is a semi-lattice in E, n

and

U H; E 9,

9 an ultra-SJ-filter in E, H; E SJ (i =

I, ... , n),

then H; E 9 for at least one i.

1

Proof. In the contrary case, by (6.1.26), E - H; E 9 for every i, thus

1

n

- H;)

=E -

U H; E 9 would 1

n

n (E -

be true.

I

6.1.

222

(6.1.28

EXTENSIONS OF TOPOLOGICAL SPACES

(6.1.28) Let SJ be a semi-lattice in E, 91 a ~-filter in E, and 1)2 an ultra-SJ-filter. If 0 ~ 91 (n) 92, then 91 < 92· Thus if both 91 and 92are ultra-SJ-filters and 91 =I= i= 92, then there exist H1 E h1 n SJ, H2E 92n SJ such that H1 n H2 = 0. Proof. 9 = 91 (n) 92 is a SJ-filter by (2.1.24) and (2.1.17), and 9 ::J 92so that

9 = 92· I (6.1.29) Let SJ be a semi-lattice in E. Then every SJ-filter in in an ultra-SJ-filter in E. Proof. Similarly to the proof of theorem (5.2.15) we have system of SJ-filters in Eis inductive. However if {9;: i EI} is a in E ordered with respect to inclusion, then 9 = U 9; is a

a SJ-filter.

I

E can be included

to prove that the system of SJ-filters filter, and clearly

iEI

Now let S:f3 be a lattice consisting of subsets of E and 0, EE S:fs, i.e. (6.1.30) (a) PE S:f3 implies Pc E; (b) 0, EE S:fs; (c) Pi, P 2 E S:f3 implies P1 u P 2 E \:fs, P 1 n P 2 E S:fs, and let O consist of the complements of sets belonging to \:fs, We of the sets having the form P n Q where PE S:fs, Q E 0: (d) 0 = {E - P: PE S:fs}; (e) We = \:fs (n) 0. Then O is evidently a lattice, We a semi-lattice and \:fs c We, 0 c We. I (6.1.31) Under the conditions (6.1.30), let x EE. The sets from We containing x constitute a filter base which generates an ultra-We-filter in E. Proof. The fact that the sets in We containing x constitute a filter base is obvious. Let m be the filter in E generated by it. If ME We, M = P n Q, PE S:fs, Q E 0, then either x E M and ME m, or x EE - M = (E - P) u (E - Q), and then either x EE - P or x EE - Q; since E - P E O c We, E - Q E S:f3 c We, therefore either E - P E m, or E - Q E m, and in any case E - ME m. Thus on account of (6.1.26) m is an ultra-We-filter. I

The ultra-We-filter described in (6.1.31) is said to be the trivial ultra-We-filter belonging to the point x. The ultra-We-filters are in a simple connection with the ultra-1:fs- and ultra0-filters: (6.1.32) Under the conditions (6.1.30), the ultra-1:fs- ( ultra-0-) filters are nothing else but those ultra-We-filters which are at the same time \:fs- (0-) filters. Proof. Since S:f3 c We; 0 c We, therefore every S:fs- or 0-filter is at the same time an We-filter too. Therefore a S:fs- (or 0-) filter m which is an ultra-We-filter is at the same time an ultra-1:fs- (ultra-0-) filter; for,. if & ::J m is a S:fs- (0-) filter, then & is an We-filter, thus & = m. On the other hand, if & is an ultra-1:fs- (ultra-0-) filter, then it is an We-filter too, moreover by (6.1.26) an ultra-We-filter. In fact, if M = P n Q, PE S:fs, Q E 0,

(6.1.37)

6.1.d.

223 ·.

,\;)-FILTERS,

then applying (6;1.26) for the semi-lattice $ (0) one of PE~ and E - PE 0 as well as one of Q EO and.E - Q E $ belongs to 5; PE&, Q E &implies ME&, and if. E - P E & or E - Q E & then a fortiori E :.__ M = (E - P) u (E - Q) E E &· I (6.1.33) If, under the conditions (6.1.30), t is a filter base in E, then the sets A E such that there can be found PE $ and R E t for which A :::, P :::, R constitute a $-filter denoted by $(t). Proof. If Ai, A 2 E $(t), then

c

A1 :::, P 1 :::, R1, A2

:::,

P2

:::,

R 2 , P 1, P 2 E $; Ri, R 2 E t

and ·if RE t. is such that R c R1 n R.2, then A1n A 2

:::,

P1 n P 2

:::,

R, P 1 n P 2 E $, RE t.

Since $(t) is evidently an ascending system in E, it is a filter in E and clearly a $-filter. I · (6.1.34) Under the hypotheses of (6.1.33), $(t} is the finest $-filter coarser than t.1 (6.1.35) Under the conditions (6.1.30), let ti, t 2 be filter bases in E, t 1 < t 2 • Then '$(t1) < $(t2), t 1 ry t 2 implies $(ti) = $(t2). I (6.1.36) Under the conditions (6.1.30), let m1 and m 2 be ultra-SJR-jilters, m 1 =I= =I= m2 • Then (a} $(m1) =I= $(mz); (b) If m1 is a 0-filter, then 0 E m1 (n) $(m 2). Proof. According to (6.1.28), there are M 1 E mi, M 2 E m 2, Mi, M 2 E SJR such that

M 1 nM2 = 0. (a): M 1 = P n Q, PE $, Q E 0. Evidently PE mi, thus PE $(mi), and if in contradiction to our statement ~(m1) = ~(m 2) were true, then PE $(m2) c m 2 would hold as well. On the other hand, if E - Q E m2, then by E - Q E $, E - Q E $(m 2) = $(m1) c m1 would hold, although E - Q ~ m1 by Q E rtt 1. Therefore E :.__ Q ~ m 2 and by (6.1.26) Q E m 2. Finally M 1 E m 2, although M 1 n n M 2 = 0, M 2 Em2. (b): Now we can suppose that M 1 E 0, thus E - M 1 E $, E - M1 :::, M 2 E m2 implies E - M1 E $(m2). I In the following it will be supposed that a topology Sf is given on the set E and \,ls is a base for Sf. (6.L37) If, under the conditions (6.1.30), ~ is a base for the topology Sf and m is the trivial ultra-SJR,-filter belonging to the point x EE, then l,ls(m) = IJ(x) where IJ(x) is the Sf-neighbourhood filter of the point x. · Proof. If VE l,ls(m), then

XE MC p C V, ME SJR, p

E$,

224

6.1.

EXTENSIONS OF TOPOLOGICAL SPACES

(6.1.38)

and, as P is open, VE b(x). Conversely, if VE b(x), \,ls being a base, there is a p E \,ls C such that XE p C V, thus VE l,J3(m). I

m

(6.1.38) ff, under the conditions (6.1.30), \,ls is a base of ff and m is an ultra-Wcfilter, x EE, then the following statements are equivalent: (a) m ~ x; (b) x is a cluster point of m. Proof. (a) => (b): (5.2.24). (b) => (a): x EP E ~· implies by (6.1.26) PE m. I (6.1.39) If, under the conditions (6.1.30), \,ls is a base for ff and t ~ x, then l,ls(t) ~ x. Proof. If x E PE \,ls, then there exists an R E t such that R c P, thus PE l,ls(t). I

An important case is that where ff is a given topology on E and \,ls coin,zides with the system of all ff-open sets. Then the conditions of (6.1.30) are evidently fulfilled, 0 will be the system of all ff-closed sets, We will consist of the sets of the form G n F where G is open and Fis closed and l,ls(t) defined in (6.1.33) will evidently consist of all neighbourhoods of all sets R E t. The latter is usually called the neighbourhood filter of the filter base t and denoted by b(t). For this case, the following further statements can be made: (6.1.40) Let [X, ff1 ] and [Y, ff 2 ] be two topological spaces,!: X ~ Y continuous, m an ultra-Wei-filter in X and & the filter generated in Y by the filter base f(m). Then M 2 E lffi2 implies either M 2 E iJ or Y - M 2 E &· Therefore if & is an illcdi,lter, then it is an ultra-'ifJ'c2-jilter. ('ifJ'c1 and 'ifJ'c2 denote the semi-lattices consisting of the intersections of open and closed sets with respect to the topologies ff1 and ff2 respectively.) Proof. Let M 2 E 'ifJ'c2 , M 2 = F2 n G2 , F 2 ffz-closed, G2 W2-open. ThenJ-1(M2) = = 1-1 (FJ nJ- 1 (G2) E'ifJ'ci, because 1- 1(F2) is ffrclosed, 1-1 (G2) is Wi-open. Thus by (6.1.26) either J-1(MJ Em or X - J-1 (M2) = J-1 (Y - M 2) Em. In the first case f(J- 1(MJ) c M 2 implies that M 2 , in the second one f(J- 1(Y - M 2 )) c c Y - M 2 implies that Y - M 2 belongs to 5. Hence if 5 is an 'ifJ'c 2-filter, then it is by (6.1.26) an ultra-lffi2-filter. I (6.1.41) Let [X, ffi] and [Y, &r2 ] be two topological spaces,/: X ~ Y continuous, t a filter base in X. Then blf(t)) is a (8i', 8i~ I/(E'))-homeomorphism; furthermore, it can be seen from (6.1.19) that this is the only homeomorphism fixing E and mapping [E', 8f'] onto a subspace of [£~, 8i~]. On account of (6.1.46) and (6.1.48) h(E') => E{ holds iff 8i~ I h(E') is compact, i.e. iff 8i' is compact. According to (6.1.47), if h(E') = E{, then {x} is 8f'-closed for the points x EE' - E. Let us suppose now that 8i' is compact and {x} is 8i'-closed whenever x EE' - E. In this case h(E') => E{ according to the preceding, thus we still have to show that m'(x) is a 0-filter whenever x EE' - E. If x has this property and SE m'(x), then there is an Mc S such that M = P n Q, PE $, Q E 0, x E E s'(P) n Q. As {x} is closed, for every pointy EE' - s'(P), there exists a Py E $ such that y E s'(Py), x ~ s'(Py). The 8i'-closedness of E' - s'(P) implies that it is 8i' -compact, hence n

E' - s'(P) c

LJ

s'(Pn),

1

and, with the notation n

Qo = Q n

n (E -

Py) E 0,

1

on account of (6.1.9), xEE - PYI (i = I, ... , n) and, by (c), x E Q0 , Q0 E m'(x), finally n

E - P = En (E' - s'(P)) C En

LJ

n

s'(Py1)

1

= LJ PYI 1

implies P => E -

so that S => M

n

n

1

1

u PYI = n CE -

Py,),

= P n Q => Qo EO n m '(x). I

The following theorem introduces an important type of compactification: (6.1.50) Let [E, 8i] be an arbitrary topological space, $ a lattice which is a base for 8i, 0, EE$, 0 = {E - P: PE$}. Then there exists a space [E', 8i'] with the following properties: (a) 8i' is an extension of 8i;

(6.1.52)

6.1.e.

WALLMAN-TYPE COMPACTIFICATIONS

229

(b) The sets Q (where Q E O and the closure is taken with respect to 8!') constitute a closed base for Sf'; --(c) Qi, Q2 E O implies Q 1 n Q2 = Q1 n Q2 ; (d) Sf' is compact; (e) x EE' - E implies that {x} is Sf' -closed. Each space [E', Sf'] with these properties is said to be a Wallman-type compactification of [E, Sf] ( and Sf' one of Sf) corresponding to \,ls. Two spaces with properties (a)-(e) can be mapped onto each other by a uniquely determined homeomorphism fixing E. A space [E', Sf'] of this kind satisfies also the following conditions: (f) x EE' - E implies that {x} is the intersection of suitable sets of the form Q (Q E0); (g) If x EE' - E and Vis a 8!'-neighbourhood of x, then there is a Q E Osuch that x E Q c V. Proof. The existence of a space with properties (a)-(e) is contained in (6.1.44), (6.1.46), and (6.1.47), and the uniqueness follows from (6.1.49). (b) and (e) clearly imply condition (f). Suppose now that (a), (c), (d), (e), and (f) are fulfilled. Let Vbe a Sf'-open set containing x EE' - E. Consider those Q (Q E 0) for which x E Q. By (c), they constitute a filter base. If each Q in question intersected the set E' - V which is compact by (d), then their intersection would meet E' - Vas weH, in contradiction to (f). Hence there is some Q EO such that x E Q c V. I

On account of (6.1.16): (6.1.51) If, under the hypotheses of (6.1.50), [E, Sf] is a T0-space, then its Wallman-type compactification belonging to \,ls is a T0-space as well. I (6.1.52) The Wallman-type compactification of [E, 8!] belonging to \,ls is an Sispace if! (a) for x E PE \,ls, there is a Q E O such that x E Q C P, i.e. if! (b) for x EE, the trivial ultra-lJR-filter belonging to x is a 0-filter; under the hypotheses of (6.1.50), of course O = {E - P: PE \,ls}, ~(x) < 5(x) < m(x) (where v;(x) denotes the sr;-neighbourhood filter of x) g2 -+ x with respect to 81"~. On account of g < g1 < g2, x must be in both cases a 81"~-cluster point of g. I

Concerning the separation properties of [E~, 81"~] the following can be said: (6.1.68) With the notations of (6.1.67), x EE~ - E, y EE~, x =fay imply that x and y are disconnected. Therefore if [E, 81"] is a T0-, S 1-, Tr, S 2- or T 2-space, then fE~, §~] has the same property. Proof. As m(x) is an open filter whenever x EE; - E, we have by (6.1.34) 5(x) = t>(m(x)) = m(x). Consequently, if x, y EE; - E, x =fa y, by m(x) =fa m(y) and (6.1.28) 0 E m(x) (n) m(y), and with this 0 E 5(x) (n) 5(y) is fulfilled. On the other hand, if x E E, y EE~ - E, then 0 ~ 5(x) (n) 5(y) would imply, again by (6.1.28), that 5(y) > 5(x) = b(x), i.e. m(y) = 5(y) is convergent with respect to 81". Thus the statement follows from (6.1.14). I

We can obtain from theorems (6.1.64), (6.1.67), and (6.1.68):

(6.1.70)

6.1.i.

EXERCISES

241

(6.1.69) If [E, 8f] is a T 2-space, then the space [~, 8f~] occurring in (6.1.67) is a H-closed extension of[E, 8f]. I On account of this, (6.1.64} can be completed as follows: (6.1.70) A T 2-space is H-closed iff it is almost compact. Proof. If the T2-space [E, 8f] is almost compact, then it is H-closed by (6.1.64). On the other hand, if this space is not almost compact, then, by (6.1.66), there is a non-convergent ultraopen filter in it. Therefore the extension constructed in (6.1.67) is a proper T2-extension; thus [E, 8f] is not H-closed. I 6.1.i. Exercises, 1. Let [E', 8f'] be a topological space in which the interiors of closed sets constitute a base, E c E' dense, gr = gr, I E. Show that (a) the closures of gr,_open sets constitute a closed base with respect to 8f'; (b) if G' is gr'-open, then G' = G' n E; (c) gr' is a strict extension of Sf.

2. Show that, in a regular space, the interiors of closed sets constitute a base· On the other hand, let E = A u B u Cu {a} u {b}, where the sets on the righthand side are disjoint, further A = {au: i,j EN}, B = {bu: i, j EN}, C = {ci: i EN}, V(au) = {au}, V(b;) = {bu}, and for n EN Vn(ci) = {c;} u {au: j ~ n} u {bij: j ~ n}, Vn(a) = {a}u{au: i~n,jEN}, Vn(b) = {b}u{bu: i~n,jEN}. Show that (a) the neighbourhood bases b(au) = {V(au)}, b(bu) = {V(bu)}, b(c1) = {VnCci): n EN}, o(a) = {Vn(a): n EN}, o(b) = {Vn(b): n EN} define a topology gr on E; (b) gr is a T2-topology; (c) V(aij), V(b1i), Vn(ci) are open-closed; (d) Vn(a) = int Vn(a), Vn(b) = int Vn(b); (e) Vn(a) - Vm(a) =/.= 0, (n, m EN); (f) the interiors of closed sets constitute a base for gr, but gr is not regular.

3. Let E = R - Q, E' = R, E" = Ru {co} where co~ R, 8f = $IE, 3!' = $, ~ is the filter in R generated by the sets having the form (x, + oo) - N (x ER), and 8f" be that strict extension of gr, which is obtained by choosing Meo) = ~Show that (a) 3f' is a strict extension of 8f, and 8f" one of 8''; (b) [E", 8f"] is a T2-space; (c) N is 3!"-closed; (d) if A c E, and the 8f"-closure of A contains N, then it contains co as well; (e) 3!" is not a strict extension of 3!. 4. Let [E, gr] be a topological space, E' :::> E, and let us assign to every x EE' 16 Akos Csaszar

242

6.1.

EXTENSIONS OF TOPOLOGICAL SPACES

an open filter 5(x) in E, in particular, if x EE, let 5(x) = tJ(x) be the 8f-neighbourhood filter of x. Let o'(x) = {Su {x}: SE 5(x)}. Show that (a) o'(x) is a neighbourhood base of x for a topology 8f'; (b) 8f' is a reduced extension of 8!; (c) 5(x) is the trace filter in E of the 8f'-neighbourhood filter tJ'(x) of x EE', (d) if§" is a topology on E' furnishing the trace filters &(x), then 8!" < §'. 5. Under the hypotheses and with the notations of 4, show that (a) 8f' is an Si-topology iff 8f is an Si-topology and x EE' - E implies

n {S:

SE 5(x)} = 0;

(b) 8f' is an S 2-topology iff 8f is an S 2-topology, &(x) has no 8f-cluster point whenever x EE' - E, and x EE' - E, y EE' - E, x =I= y imply 0 E 6(x) (n) 5(y). 6. Under the hypotheses and with the notations of 4, let §" be the strict extension of § belonging to the trace filters &(x). (a) Show that if 8!' is an Srtopology, then 8!" has the same property; (b) Give an example for the case when 8f' is a Ti-topology without 8f" being reduced; (c) Give an example for the case when§' is a Ti-topology and 8f" is a T0-topology but not an S1 -topology. 7. Show that the m-dili1ensional sphere

is homeomorphic with the Alexandroff compactification of the space [Rm, ~m ]. [Denoting the latter by [E, §], where E = Rm u {w }, the map given by the formulas f(Xi , · · · • , Xm) = (

2xi r+l 2

' · · · ,

2xm r+l 2

'

r2 - I ) r+I

-2--

'

m

r 2 = 'f.x;,f(w) = (0, ... ,0, l)ESm 1

the projection of Rm onto Sm from the point (0, ... , 0, 1) is the (8f, ~m+J [ I Sm)-homeomorphism required.] 8. Let E = R, ~ = {[a, b): a, b ER, a~ b }. Show that ~ is a semi-lattice and give all ultra-~-filters. [The latter are generated by the filter bases r+(a) = {[a, a+ a): a> OJ and r-(a) = {[a - a, a): a> O} (a ER)]. ·9. Let Ebe an infinite set, let 5,l3 consist of E and the finite subsets of E, and let 0 = {E - P: PE S,ls}, ill1 = 5,l3 (n) 0. Determine all ultra-We-filters. [These are the filters :x: (x EE) and the filter consisting of all sets of E having finite complements.]

6. l.i.

243

EXERCISES

n

10. Let E = R and let

~

consist of the sets having the form

U

(a;, b;) (n EN,

1

a;, h; ER, a; < b;) and further of 0 and E. Show that ~ is a lattice and, with the notations O = {E - P: PE ~ }, We = ~ (n) 0, give all ultra-We-filters. [The latter are generated by the filter bases r( oo) = {( - CXl, -a) u (a, + CXl): a> O}, r(a) ={a}, r+(a) = {(a, a+ s): e >0} and r-(a) ={(a - e, a): e >0} (a ER).] 11. Let 811 and 812 be two topologies on E, ~ 1 and ~ 2 base-lattices for 811 and 812 respectively, and, starting from them, construct the sets E~ and E; according to (6.1.43) and denote by 81{ and §~ the strict extensions described in (6.1.44) of 811 and 812 respectively. Show that, if ~ 1 c l,{s 2, then - with the usual notations Oi, 02, Wei, Wc 2, mi(x), mix) - : (a) In any ultra-W1 2-filter, the sets belonging to IJJc1 generate an ultra-Wcrfilter; (b) If f: E; ---+ E~ is a map fixing E such that m 1 (f(x)) is generated precisely by the sets belonging to Wc1 and contained in m 2 (x), then f is a (81~, 81{)-continuous surjection. 12. Consider the set E in 9, and the lattice ~ which is a base for the topology ®E· Construct the Wallman-type compactification with respect to \,ls of the space [E, ®E]. [E' =Eu {OJ}, the points of Eare 81'-open, the only 81'-neighbourhood of OJ is E'.] 13. The same problem for E and ~ in 10 (in which case 8J = &). [E' =Eu {OJ}, the &-neighbourhood base of x EE is a 81'-neighbourhood base as well, the only neighbourhood of OJ is E'.] 14. Let [E, 81] be a locally compact, non-compact S2-space and let ~ consist of those open sets for which either P or E - Pis compact. Show that (a) \,ls is a base-lattice ; (b) The Alexandroff compactification of [E, 81] fulfils the conditions given in (6.1.53); thus it is precisely the Wallman-type compactification of [E, 81] with respect to ~. 15. Let [E, 81] be a zero-dimensional space, let~ consist of the open-closed sets, and show that (a) if A and Bare closed sets with compact boundary and An B = 0, then there are open-closed sets C and D such that Ac C, B c D, C n D = 0; (b) if [E',81'] is the Wallman-type compactification of [E, 81] with respect to~, then 8J is an S2-topology; (c) [E', 81'] is at the same time the Freudenthal compactification of [E, 81]. 16. Let [E, 81] be a topological space, ~abase-lattice for 81, [E', 81'] the Wallman-type compactification with respect to ~- Show that (with the usual notations) every ultra-We-filter in E is 81'-convergent. 17. Let [E, 81] be a topological space, ~ a base-lattice for 81, 0 = {E - P: PE~}, [E',81'] a reduced compactification of [E,81] in which {Q:QEO} is a closed base, Qi, Q2 E :\)implies Q1 n Q2 = Q1 n Q2 and if E c Kc E' is compact, then K = E'. Show that in this case 81' is precisely the Wallman-type compactification of 8J with respect to ~16*

244

6.1.

EXTENSIONS OF TOPOLOGICAL SPACES

(6.2.1)

18. Show that a continuous image of an almost compact space is almost compact as well. n

19. Show that if [E, gr] is a topological space, E

= U E; and 8i IE; is almost 1

compact (i = 1, ... , n) then gr is almost compact as well. 20. Let [E,gr] be a compact S 2-space, aEE, {a} gr_closed, A =E- {a}, a E A. Show that [E, gr] is the Alexandroff compactification of the subspace [A, gr I A]. [On account of (6.1.11) gr is a strict extension of gr I A. If G is a gr_open neighbourhood of a, then E - G is 3f I A-compact and gr I A-closed. On the other hand, if Kc A is compact and 3f I A-closed, then it is gr_closed as well, since otherwise b(a) would have a cluster point in K by a EK which contradicts to (S 2); hence E - K is a neighbourhood of a.] 21. Let [E, gr] be the non-regular gr2 -space defined on p. 97. Show that the subspace [E0 , gr I E 0 ] is fl-closed without being compact if E 0 = [ -1, 1]. [Consider (6.1.70) and (6.1.66) (c).] 22. With the notations of the preceding exercise, show that the subspace [Ei, 3f I Ei] of the fl-closed space [E0 , gr I E 0 ] is closed without being fl-closed if E 1 = { : : n E

NJ .

.

23. Show that every almost compact space is pseudo-compact. 24. Show that the space [W, gr] in exercise 5.3.g.13 is countably compact (and pseudo-compact) without being almost compact. [(6.1.65).] 25. Let ~ be a base-lattice in a topological space [E, gr], and suppose that conditions (6.1.50) (a), (c), (d), (e), (g), (6.1.52) (a), and (6.1.53) (c) are satisfied for an Si-space [E', &f']. As a partial sharpening of (6.1.53), show that gr' is a Wallmantype compactification of 3f corresponding to ~[For x EE' - E, let q(x) denote the system of all Q E O such that x E Q. By (6.1.50) (c), this is a filter base and, as a result of (6.1.50) (g), it generates in E an ultra-0-filter m(x). From (S1) we obtain that m(x) is non-trivial and, by (6.1.50) (e), x, y EE' - E, x # y implies m(x) # m(y). By (6.1.50) (d), each non-trivial ultra-0-filter coincides with m(x) for some x EE' - E. Hence E' can be identified with the basic set of a Wallman-type compactification corresponding to ~, having the topology &f". By (6.1.45), every Q E O has the same closure with respect to &r" as with respect to gr'. Therefore &f" is coarser than gr'_ Conversely, if Fis &f' -closed, then it is &f" -compact and F n E is gr_closed; hence if x EE' - F, y E F, then x and y are weakly gr"_separated and (6.1.52) (a) and (6.1.53) (c) imply that they are gr"_disconnected so that x and F have disjoint &f"-neighbourhoods and gr" is finer than gr'.] 26. Let [E, gr] be a compact S 2-space and ~ a base for gr which is a lattice, 0, EE~- Show that~ satisfies conditions (6.1.52) (a) and (6.1.53) (c). [Either from (6.1.50) and (6.1.53) or directly using (S 2) and the compactness of E-PforPE~.]

(6.2.2)

6.2.a.

EXTENSION OF CONTINUOUS MAPPINGS

245

27. Let [E', §'] be a compact Trspace and ~,abase for 8f' such that ~, is a lattice and 0, E' E ~'. Suppose that P' = int P' for P' E ~'. Show that if E c E' is 81'-dense and 8f = §' IE, then ~ = ~, (n) {E} is a base-laltice for 8T satisfying conditions (6.1.52) (a) and (6.1.53) (c), and [E', 81'] is a Wallman-type compactification of [E, §] corresponding to ~. [Q' = E' - P', P' E ~' implies Q' = int Q', hence Q' = Q' n E. Hence Q; = =E' - P;,P; E ~', (i = 1, 2), Q' = Q~ n Q;, (Q{ n E) n (Q; n E) = 0 imply Q{ n En n Q; n E = 0. Therefore (6.1.50) (a), (b), (d), (e), and (6.1.53) (c') are fulfilled for ~- Finally apply (6.1.52) and (6.1.53).] 28. Show that an S2-space is rim-compact iff it has a compactification with a base such that the boundaries of the elements of the base are contained in the given space. 6.2 EXTENSION OF MAPPINGS 6.2.a. Extension of continuous mappings, In connection with the question of the extension of topological spaces studied above, the following problem arises quite naturally. Let [X, §] be a topological space, [X', 8f'] an extension, [Y, &r*] a further topological space, f: X -+ Ya (&r, 8f*)-continuous map. The question can be raised whether there exists a continuous extension off onto X', i.e. a map g: X'-+ Y which is (§"', §"*)-continuous and for which g IX= f A necessary condition for the existence of such a g can be formulated at once. For this purpose, denote as usual the 8f-neighbourhood filter of x EX' by b(x), its trace filter in X by b'(x) (n) {X} = &(x). Since &(x) > b'(x), &(x) -+ x with respect to §"', thus if g is (§', §*)-continuous, then by (2.6.13) g(&(x))-+ g(x) and g I X = f impliesf(&(x)) -+ g(x). On account of this and of (2.6.22), we have:

(6.2.1) Let [X, §] and [Y, 81*] be two topological spaces, [X', §'] an extension of the space [X, §], f: X-+ Y, b'(x) the §'-neighbourhood filter of x for x EX', &(x) = b'(x) (n) {X}. In order that a (§', §*)-continuous map g: X' -+ Y exists for which g I X = f, it is necessary that f be(§, §*)-continuous andf(&(x)) a W*-convergent filter for each x EX'. I

It is an important fact that, if §* is regular, these conditions are also sufficient for the existence of a continuous extension: (6.2.2) With the notations of (6.2.1), let§* be regular,!(§", §*)-continuous, and suppose that, if x EX', then f(&(x) is a §*-convergent filter. For x EX, let g(x) = = f(x), and if x EX' - X, choose the point g(x) E Y such that f(&(x)) -+ g(x) with respect to §*. Then g is (§', W*)-continuous, and g IX= f Proof. Since the map f is (8f, 8f*)-continuous, f(&(x)) -+ f(x) = g(x) holds for x EX too. By (2.6.10), it is to be shown that x EX' implies g(b'(x)) -+ g(x). Let V* be an arbitrary 8f*-neighbourhood of g(x), and Vt c V* a closed §*-neighbour-

6.2.

246

EXTENSION OF MAPPINGS

(6.2.3)

.

hood of g(x); By f(&(x)) -+ g(x), x has a 8f '-open neighbourhood G such that /(G n X) c Vt. For an arbitrary pointy EG, G is a 8f'-neighbourhood of y as well, hence G n XE &(y), and, as a consequence of /(&(y)) -+ g(y) and (5.2.24), g(y) E E/(G n X) c Vt c V*. Accordingly g(G) C V*. I Concerning the uniqueness of the continuous extension, the following can be said: (6.2.3) With the notation of(6.2.l), let u1:: X'-+ Yandg 2 : X'-+ Y be (8'', 8'*)continuous, 8'* a T 2-topology and g 1 I X = g 2 I X = f Then g 1 = g 2 • Proof. For an arbitrary point x EX', by &(x)-+ x and (2.6.13), g 1(&(x))-+ g 1(x), gi&(x)) -+ g 2(x), and U1(&(x)) = g 2(&(x)) = f(&(x)). By (2.5.19), therefore g 1(x) =

= ulx).

I

As a first application of theorem (6.2.2), an important property of the Wallman compactification will be shown: (6.2.4) Let [X, 8'] be an arbitrary topological space, [X', 8''] its Wallman compactffication, [Y, 8'*] a compact S2 -space, f: X-+ Y (8', 8'*)-continuous. Then there exists a (8f', 8'*)-continuous extension off Proof, On account of (6.2.2), it is to be shown that f(&(x)) is a 8'*-convergent filter for x EX', because 8'* is regular, according to (5.3.22). However, with the usual notation, &(x) = tJ(m(x)), where m(x) is an ultra-Wc:-filter in X. As 8'* is compact,· the filter base m* generated in Y by f(m(x)) has a cluster point y E Y. If G is an open neighbourhood of y, then by (6.1 .40) eithe~ G Em * or Y c.... G Em *. The latter contradicts the fact that y is a cluster point of m*, thus GE m* and therefore m* -+ y. But by (6.1.39) tJ*(m*) -+ y also holds, and on account of (6.1.35) and (6.1.41) tJ*(m*) = tJ*(f(m(x))) < /(tJ(m(x))) = f(&(x)). Thus f(&(x)) -+ -+ Y-1 Notice that the proof of (6.2.4) furnishes a much :rp.ore general result. In fact, let ~ be the system of all open sets, and thus D, thl;tt of all .closed. sets of a topological space [X, 8f], while [X~, 8'~] is the extension of [X, 8f] constructed according to (6.1.43) and (6.l.44) (now putting X and X~ instead of E and E~ respectively). Now, if [Y, 8f*] is a compact S 2-space and/: X-+ Y is (Sf, 8'*)-continuous, then, according to the proof of (6.2.4), it can be seen that there exists a (8'~, 8f*)-continuous extension off From here it follows at once that if X c X' c X~ and 8f' = = 8'~ X', then the restriction to X' of the (Sf~, 8f*)-continuous extension off furnishes a (8f', §*)-continuous extension off Now we can go back to the statement of (6.2.4) choosing for [X', Sf'] the Wallman compactification, while as a further application [X', Sf'] can be chosen, e.g. as the H-closed extension of the space [X, 8'] constructed in· (6.1.69). J

6.2.b.· Extension of uniformly continuous mappings. Now let [X, 611] and [Y, 611*] be two uniform spaces,!: X ~ Ya (611, 611*)-uniformly continuous map and examine the question whether f can be exten.ded in a uniformly continuous way

(6.2.7)

6.2.b.

EXTENSION OF UNIFORMLY CONTINUOUS MAPPINGS

247

to an extension [X', 61.t'] of the space [X, 61.t], i.e. whether there can be found a (61.t', 61.t*)-uniformly continuous map g: X' ~ Y for which g [ X = f Of course we understand by the fact that the uniform space [X', 61.t'] is an extension of the uniform space [X, 61.t] (or that 61.t' is an extension of 61.t) that X c X', 61.t' [ X = 61.t, and Xis 8J'tt,-dense in X'. A necessary condition for the existence of such a g is, by (3.2.50), that/has a (SJ''tt,, 8J'tt•)-continuous extension to X'. It is an important fact that this condition also is sufficient: (6.2.5) Let [X, 61.t], [X', 61.t'], [Y, 61.t*] be three uniform spaces, [X', 61£'] an extension of [X, 61£], /: X ~ Y (61.t, 61£*)-uniformly continuous, g: X' ~ Y (8J'tt,, SJ''tt•)-continuous and g [ X =f Then g is (61£', "U*)-uniformly continuous as well. Proof. Let U* E 61£* be a given surrounding, 61£! E 61£* a surrounding such that Uf o Uf o Uf c U*, U E 6U a surrounding such that (referrmg to the uniform continuity off) (x, y) E U implies (f(x),f(y)) E Uf, further U' E 6U' a surrounding such that U' n (E xE) c U, finally U{ E 61.t' a surrounding such that U{ o U{ o o U{ c U'. It will .be shown that (x, y) E U{ implies (g(x), g(y)) E U*. Denote by o'(x) the 8J'tt,-neighbourhood filter of the point x EX'. Since g is (8f'tt,, 8J'tt•)-continuous, there is for U'f(g(x)) which is a 8i'tt.-neighbourhood of g(x) a V{ E o'(x) such that g(V{) c Uf(g(x)). Similarly there is a V~ E o'(y) such that g(V~) c Uf(g(y)). As V{ n U{(x) E o'(x) and Xis 8f'tt,-dense, we can find a point x 1 E V{ n U{(x) n X. In the same way we see that there is a point Yi E V~ n n U{(y) n X. Now (x1 , x) E U{,(x,y) E U{, (y, y 1) E U{ imply (x1 , Yi) E U', then (x1, Yi) E U, and hence (J(x1), /(Yi)) = (g(x1), g(Ji)) E Uf. Since g(x1) E.Uf(g(x)), g(yi) E Uf(g(y)), finally(g(x),g(y)) EU*. I For the sake of the next proof, let us note the following remark: (6.2.6) lf [E', 61.t'] is an extension of the uniform space [E, 61£], then the topology 8J'tt, is a ( strict) extension of3f'tt. Proof. By hypothesis, Eis SJ''lt,-dense in E'. From· 61.t' [ E = 61£, it follows on account of (3.2.35) that SJ''lt, [ E = 3f'ii. Thus the topology 8J'lt, is an extension of 8i'lt, namely a strict extension by (6.1.11) because 8J'tt, is regular by (3.1.15) and 8i'lt, = 3l§''tt'· I Now the following important extension theorem can be proved: (6.2.7) Let [X, 61£], [X', 61.t'], [Y, 61.t*] be three.uniform spaces, [X', 61.t'] an extension of [X, 61.t], and let [Y, 61£*] he complete. If f: X ~ Y is (61£, 61.t*)-uniformly continuous, then there is a (61£', 61.t*)-uniformly continuous g: X' ~ Y such that g[X=f . , , . .· Proof. According to (6.2.5), it suffices to show that/ has a (8J'll,, 8i'll*)-continuous extension and to prove this, by (6.2.2), we must show that if 5(x)denotes the trace filter in X of the Sf'll,-neighbourhood filter of the point x EX', then /(5(x)) is 3J'lt*-convergent for every x E X'. In fact, the regularity of Sf'tt* follows by 8T''tt• ·= = 8J&'tt• from (3.1.15), the (8i'll, 8i"l:t*)-continuity off results from (3.2.50) and [X', 8J'lt, ] is ari extension of [X, 8i'll] on account of (6.2;6).

248

6.2.

EXTENSION OF MAPPINGS

(6.2.8)

Now 3(x)--+ x with respect to ST'lf', thus it is a 611'-Cauchy filter by (5.1.1) and then it is a 611-Cauchy filter as well on account of (5.1.6). Hence, by (5.l.2),f(3(x)) is a 611*-Cauchy filter and, as 611* is complete, it is ST'lf.-convergent. I 6.2.c. Extension of proximally continuous mappings. Theorems similar to the preceding ones can be proved in connection with proximally continuous maps. For this purpose, the following terminology will be introduced: the proximity space [X', 8i''] is said to be an extension of the proximity space [X, 8i'] if X c X', Si'' IX= 8i' and Xis ST&,-dense in X'. At the same time Si'' will of course be called an extension of Si'. It follows at once from the definitions:

(6.2.8) If [E', 611'] is an extension of the uniform space [E, 611], then 8i''U, is an extension of 8i''U. Proof. By (3.2.35) 8i''lf, IE = 8i''U'• and on account of ST'lf, = ST&'lf, Eis ST&'lf,dense in E'. I (6.2.9) If [E', 8i''] is an extension of the proximity space [E, 8i'], then the topology ST&, is a strict extension of ST&. Proof. By (3.1.27) ST&, IE= ST& and according to the hypothesis Eis ST@,-dense. ST&, is regular by (3.1.15), thus it is a strict extension of ST& by (6.1.11).1 The following two theorems, analogous to (6.2.5) and (6.2.7), are valid as well: (6.2.10) Let [X, 8i'], [X', 8i''], [Y, 8i'*] be three proximity spaces, [X', 8i''] an extension of [X, 8i' ],f: X --+ Y (8i', @*)-proximally continuous, g: X' --+ Y C3T@· ,ST&•)continuous, g I X = f Then g is (8i'', @*)-proximally continuous as well. Proof. Let 611' and 611* be precompact uniformities inducing the proximity relations Si'' and Si'*; these exist on account of (4.2.25). With the notation 611' I X = = 611, by (3.2.35), 8i''lf = Si', thus f is (611, 611*)-uniformly continuous by (3.2.77). Therefore (6.2.5) can be applied and shows thatg is (611', 611*)-uniformly continuous, and thus (Si'', @*)-proximally continuous on account of (3.2.50).1 (6.2.11) Let [X, 8i'], [X', 8i''], [Y, Si'*] be three proximity spaces, [X', 8i''] an extension of [X, 8i'], [Y, 8i'*] compact. If f: X--+ Y is (8i', 8i'*)-proximally continuous, then there exists a (8i'', 8i'*)-proximally continuous g: X' --+ Y for which g IX= f Proof. Introduce again the precompact uniformities 611' and 611* inducing Si'' and Si'* respectively. With the notation 611 = 611' I X,/is now (611, 611*)-uniformly continuous and 611* is complete by (5.2.22). Then (6.2.7) shows that/ possesses a (611', 611*)-continuous extension g: X'--+ Y which is also (Si'', @*)-proximally continuous by (3.2.50).1 6.2.d. Exercises. 1. Let [Ei, 8fi] and [E2 , ST2 ] be two topological spaces, ~ 1 and base-lattices for ST1 and W2, [E{, W{] and [E~, W~] the corresponding Wallmantype compactifications. Show that if ST~ is an S2-topology and/: E 1 --+ E 2 is a mapping such thatP2 E% implies/- 1 (P2) E8i'i, then/has a (ST{, ST~)-continuous extension g: E{--+ E~ (thus for which g I~~=/). ~ 2

(6.3.2)

6,2.c.

EXTENSION OF PROXIMALLY CONTINUOUS MAPPINGS

249

[If m1 is an ultra-1.lJh-filter and y is a 8f;-cluster point of f(m 1), thenf(l,ls(m1))--+y.] 2. Let [E1 , 8i"1 ] be a rim-compact space, [E2 , 8i"2 ] a zero-dimensional T0-space, [E{, §"{] and [E;,, 8f;,] the corresponding Freudenthal compactifications, f: E1 --+ E 2 (8i"i, §" 2)-continuous. Show that the1e exists a (§"~, §";)-continuous g: E{ --+ .E;, for whichg I~: =f [Use the result of exercise 6.1.i.15 as well as 1.] 3. Let \,l51 and ~ 2 be base-lattices on E for two topologies §°1 and §"2 respectively, [E{, §"{] and [E;,, 8f;,] the corresponding Wallman-type compactifications. Show that, if ~ 2 c ~ 1 , there exists a (§"{, §";,)-continuous surjectionf: E{ --+ E;, fixing E. 4. Let f be an ~-continuous function such that x, y ER implies f(x + y) = = f(x) + f(y). Show that, for a suitable c ER,f(x) = ex for every x ER. [Choosing c = f(I), f(x) = ex if x E Q.] 5. Let [X, p] and [Y, a] be pseudo-metric spaces, E c X 8fP-dense, [Y, a] complete, f: E --+ Y a map such that x1 , x 2 EE implies

a(f(x1), f(xJ)

~

c p(xi, x 2 ),

for suitable c > 0. Show that there exists a g: X --+ Y such that g I E = f, and Xi, x 2 EX implies a(g(xJ, g(x2 ))

~

c p(xi, x 2 ).

[f is ("UP I E, 611..)-uniformly continuous.] 6. Let

Ea set such that the points x EE; - E can be associated in a one-to-one manner with all round Cauchy filters non-8T'LI-convergent in [E, 6UJ. Denoting by &(x) the filter associated in this way with the point x EE; - E and making &(x) equal to the 8T'U-neighbourhoodfilter b(x) of x whenever x EE, let 6U; be the uniformity on E; constructed according to (6.3.16). Then 6U~ is complete and a reduced extension of 6U. Proof. Only the completeness of 6U; is to be proved; the rest follows from (6.3.21). Thus let r' be a 6U;-cauchy filter base. Consider the proximity filter ,p'(r') of r' with respect to the proximity &'Lie· This is a 6U;-cauchy filter by (6.3.14) and &'Ll 0-round on account of (6.3.9). By (6.3.7), we can speak of the filter & = ,p'(r') (n) {E} and by (6.3.6) & is a round filter with respect to the proximity &'LI;, I E = &'LI (this equality follows from (6.2.8)). & is a 6U;-cauchy filter base by (5.1.7) and hence a 6U-Cauchy filter on account of (5.1.6). Therefore there is an x EE; such that & = &(x); namely if &converges to a point x E Ewith respect to 8T'U, then & = &(x) by (6.3.13) while if & is not 8T'Ll-convergent, then & = &(x) will hold for some point x EE; - E. However, denoting the 8T,"W neighbourhood filter of x by b'(x), &(x) = b'(x) (n) {E} > b'(x) implies 5(x) --+ x with respect to 8T'U~, hence xis a 8T'Ll0-cluster point of the filter ,P'(r') by (5.2.24). On account of (5.2.28), ,p'(r') --+ x, and by r' > ,P'(r') a fortiori r' --+ x with respect to 8T'Ll;,· I This can be completed by the following remark: (6.3.24) Let [E. 6U] be an arbitrary uniform space, [E;, 6U;] its extension constructed in (6.3.23), E c E' c E;. If [E', 6U; IE'] is complete, then E' = E;. Proof. Suppose that x EE; - E'. Then &(x)--+ x, but &(x) is also a (6U; I E')Cauchy filter base by (5.1.6) so that there is a y EE' such that &(x) --+ y with respect to 8T'LI;, IE' = 8T'LI;, IE', i.e. with respect to 8T'LI;, (on account of (3.2.35) and (2.4.8)). However, x and y are weakly separated, thus disconnected by the regularity of 8T't!;, and this contradicts (2.5.1).1 (6.3.25) Let [E, 6U] be a uniform space, [E;, 6U:] its extension constructed in (6.3.23), and [E', 6U'] an arbitrary reduced extension of [E, 6U]. Then there exists a uniquely defined unimorphism hfixing E which maps [E', 6U'] into a subspace of [E;, 6U;]. 61,f' is complete if! h(E') = E;. Proof. If x EE', let o'(x) be the 8T't!,-neighbourhood filter of x. Its trace filter &'(x) = o'(x) (n) {E} is a round Cauchy filter in [E, 6U] by (6.3.15) and f(x)

256

6.3.

EXTENSIONS OF UNIFORM SPACES

(6.3.26)

is not 8f'Lf-convergent if x EE' - Eon account of (6.3.21), moreover x, y EE' - E, x # y imply ~'(x) # f(y). Therefore it can be given a uniquely defined bijection h: E' ~ E" onto a suitable set E c E" c E; such that h(x) = x for x EE and ~'(x) = ~(h(x)) for x EE' - E where ~(h(x)) denotes the filter belonging to the point h(x) EE; according to (6.3.23). Denote by 611" the extension of 611 constructed on the set E" by means of (6.3.16) starting from the trace filters ~(y) (y EE"). In this case 8f'lf" and 8f'lf, are by (6.2.6) strict extensions of 8f'lf with respect to the trace filters t(y) and f (y) respectively, so that by (6.1.19) h is a (8f'lf,, 8f'Lf.)-homeomorphism. Since h IE: E ~ E" is the canonical injection of Einto E" and 611 = 611" IE, h IE is (611, 611")-uniformly continuous, thus (6.2.5) can be applied and shows that his (611', 611")-uniformly continuous. Interchanging the roles of E' and E", we get in the same way that h- 1 is (611", 611')-uniformly continuous. Therefore his the required (611', 611")-unimorphism since by (6.3.20) 611" = 611~ IE'. The uniqueness of h results from the fact that a (611', 611; / E")unimorphism is a (8f'lf,, 8f'lfc I £")-homeomorphism if E c E" c E;; thus (6.1.19) can be applied. On account of (5.1.12), 611' and 611; I h(E') are simultaneously complete, namely by (6.3.24) iff h(E') = E;. I The uniform space [E', 611'] is called a completion of the uniform space [E, 611] (and 611' one of the uniformity 611) if 611' is a reduced complete extension of 611.

It follows from (6.3.23) and (6.3.25) that: (6.3.26) Every uniform space [E, 611] has a completion: the space [E;, 611;] constructed in (6.3.23) is of this kind. Two completions of the space [E, 611] can be mapped onto each other by .means of a uniquely defined unimorphism fixing E. I (6.3.27) Let [E, 611] be an arbitrary uniform space, [E', 611'] its reduced extension and [E", 611"] a completion of [E', 611']. Then [E", 611"] is a completion of the space [E, 611] as well. Proof. It need only be proved that [E", 611"] is a reduced extension of [E, 611] which follows from (6.2.6) and (6.1.18).1 (6.3.28) If the uniform space [E", 611"] is a completion of [E, 611] and E c E' c E", 611' = 611" IE', then [E", 611"] is a completion of [E', 611'] as well. Proof. (6.2.6) and (6.1.18) can be applied again. I We get at once from (6.3.22): (6.3.29) The completion of a separated uniform space is separated as well.

I

By means of (6.3.29), it follows from (6.3.19) that: (6.3.30) The completion of a pseudo-metrizable (metrizable) uniform space is pseudo-metrizable (metrizable) as well, I

(6.3.31)

6.3.d.

257

EXERCISES

The meaning of the term "precompact" is illustrated by the following theorem: (6.3.31) Let [E, 6U] be a uniform space. The following statements are equivalent: (a) [E, 6U] is precompact; (b) The completion of [E, 6U] is compact; (c) [E, 6U] has a compact extension. Proof. (a)= (b): If [E', 6U'] denotes the completion of [E, 6U], then by (3.2.76) 6U' is precompact as well, hence by (5.2.20) compact. (b) = (c): Evident. (c) = (a): If [E', 6U'] is a compact extension of the space [E, 6U] then 6U' is precompact on account of (5.2.21), thus by (3.2.70) 6U is precompact as well. I

6.3.d. Exercises. 1. Let &1 and &2 be round filters in the proximity space [E, SJ']. Prove that &1 (u) &2 is a round filter as well. 2. Let t be a filter base in the proximity space [E, SJ']. Show that ,\J(t) is the finest round filter coarser than t. . 3. Let t 1 and t 2 be filter bases in the pro~1mity space [E, SJ']. Show that (a) ,\J(t1 (u) t 2) = ,\J(t1) (u) ,\J(t 2); (b) ,\J(t1 (n) t 2) > ,\J(t1) (n) ,\J(t2), if 0 ~ t 1 (n) t 2 ; (c) in general, equality does not hold here. [E = R, SJ' = S!'P,' t; = {R;n: n EN}, R1n = {O} u (Q n (n, + co)), R 2n = = {O} u ((n, + co) - Q).] 4. Let [E', SJ''] be an extension of the proximity space [E, SJ'], &a round filter in [E, §']. Show that there exists a round filter&' in [E', §''] such that & = &' (n) {E}. [Let &' be the filter in E' generated by the 8f&,-closures of the sets SE&-] 5. Let }, show that &' = &q;,, [Let 61,f' be the precompact uniformity inducing&'; then 611' IE= 6114) as both sides are precompact and induce & and then we can refer to exercise 6.3.d.5.] 2. Let [E, &] be a separated compact proximity space, Ac E, A the ST@-closure of A. Show that [A, & I A] is the compactification of [A, & I A]. 3. With the above notations omit the condition that SJ' is separated and let A c A' c A be a set such that if x E A, x n A = 0, then x n A' consists of exactly one point, while x EA - A, x n A # 0 imply x 1 A'. Show that [A', & I A'] is the compactification of [A, & I A]. 4. Let [X, &] and [Y, a] be proximity spaces and/: X---+ Ya map such that if r is a &-compressed filter base in X, then /(r) is a-compressed. Show that f is (&, a)-proximally continuous. (Cf. exercise 5.1.c.5.) [Let [X', &'] and [Y', a'] be the compactifications of the given spaces, and b'(x) the §"@,-neighbourhood filter of x for x EX'. As f(b'(x)(n){x}) is compressed, it follows from (6.2.2) that f has a (81&,, STtl')-continuous extension g which is (&', a')-proximally continuous too.] 5. Let & and a be proximities on the set E and suppose that every &-compressed filter base is a-compressed too. Show that & > a. 1 8*

276

6.4.

EXTENSIONS OF PROXIMITY SPACES

6. Let

n and (xn) cannot have a gr_cluster point.] 18. Let [E, gr] be an S4-space. Show that (a) If (xn) is a sequence in E having no gr_cJuster point, then (xn) has a subf

oo

sequence (xk) such that xkn n xkm = 0 whenever n =I= m and F =

LJ xkn is closed; 1 is gr I F-continu-

(b) With the above notations, if f(x) = n for x E xk., then f ous, thus it has a gr_continuous extension; (c) If gr is pseudo-compact, then it is countably compact as well; (d) The topology gr in t.he preceding exercise is not normal. [Refer to exercise 4.2.e.1 I.] 19. Give an example of a compact T2°space which has a non-normal subspace. [The Cech-Stone compactification of [R, $] has this property.] 20. Let [E(, §''] be the Cech-Stone compactification of the completely regular space [E, gr]. Show that if E' - E consists of a single point, then §'' is identical with the Alexandroff compactification of gr_ [Rely on 8, and exercise 6.1.i.20.] 21. Let [E', "U'] be the completion of a uniform space [E, "U]. Show that [E', gr'LI,] is real-compact iff gr'LI D is real-compact for each "U-discrete subset f

DcE. [If D c Eis "U-discrete, then it is "U'-discrete, hence gr'Ll,-closed; if D' c E' is "U'-discrete, then there is a "U-discrete D c E such that D = f(D') for a suitable bijection/: D'--+ D so that the hypotheses of (6.4.41) are fulfilled.] 22. Let [E, gr] be a completely regular space. Show that the following state-

ments are equivalent: (a) Sf is separable and pseudo-metrizable; (b) for each uniformity 6U such that gr = 8T'LI, there is a pseudo-metrizable uniformity "U' < 6U with gr'LI, = gr;

, 6.4.g.

279

EXERCISES

(c) [E, 8J] has a pseudo-metrizable ordinary compactification; (d) there is a compact, pseudo-metrizable space containing [E, 8J] as a subspace. [(a)=> (b): exercise 4.2.f.22; (b) => (c): let 8J' be a proximity such that 8J = 8J1 , 61i the precompact uniformity such that 8J' = &"ll (see (4.2.20) and (4.2.26)), then a pseudo-metrizable uniformity 6\i' < 6\i with 8J"ll, = 8J is precompact by (3.2.74) and if [E", 6\i"] denotes the completion of [E, 6\i'], then [E", sr"lt.] is a pseudometrizable ordinary compactification of [E, 8J] by (6.3.26), (6.3.30), (6.3.31); (c) => (d): obvious; (d) => (a): (5.3.35), (2.4.16), (2.4.19), (2.4.15), (2.3J2).] 23. Show that the compactification of a proximity space [E, SJ'] is pseudometrizable iff there is a countable system $ of subsets of E such that A & B implies the existence of P, Q E $ with A c P, B c Q, P i Q. [If the compactification [E', &'] is pseudo-metrizable, then consider a countable base ~Hor 8J' = 8J1 ,, (see (5.3.35) and (2.4.16)) and let$ consist of the finite unions of the sets B n E (BE~); A& B implies A i' jj by (3.1.14) for the 8J'closures, then the existence of 8J'-open sets U, V such that A c U, jj c V, V n. V = = 0 (see (5.3.22)), so that by the compactness of A and Jj there are sets A;, Bj E ~ m

with n

=

A c LJ A; c U, 1

n

jj

c LJ 1

Bi

c

m

V, hence P == ( LJ A;) n EE $

(U Bi) n EE $ satisfy A c P, B c Q, P n Q

1

=

and Q

=

_

0 and P & Q by (5.3:25).

1

Conversely if the condition is fulfilled, consider the pairs (Pm Qn) (n EN) such that Pm Qn E $ and (Pn x QJ n Un = 0 for some surrounding Un from the precompact uniformity 61i inducing SY (see (4.2.26)); if the surroundings U~ are selected from 61i so that U~ o U~ c U~_ 1 n Un then {U~: n EN} generates a pseudo-metriz~le uniformity 6\i' < 6\i (see (4.2.33)); then .8J'"ll, < @"ll = & and, conversely, A SY B implies Ac Pm B c Qn for some n EN by (3.2.19) so that (A x B) n U~ = 0 and A i'Lt' B; hence &'Lt' = & and 6\i' is precompact by (3.2.74) so that (6.3.30) and (6.3.31) furnish the statement.]

7. PRODUCT AND QUOTIENT SPACES

7.1. THE PRODUCT OF TOPOLOGICAL SPACES 7 .1.a. Projective generation. It is known that if f: E - Y is an arbitrary mapping from the set E into the topological space [Y, §"], then the topology F-1(fl) is the coarsest topology §"* on E for which f is (8T*, §")-continuous. It is known as well that if

0, constituted a neighbourhood subbase for the function x ERH i.e. x: H ~ R. Denoting "the projection onto the /th factor" by p 1, we clearly have V1,. = p, 1((x(t) - e, x(t)

and a.s the intervals (x(t) - e, x(t) x(t), it can be said by (7.1,22):

+ e)

+ e)),

constitute an $-neighbourhood base of

(7.1.25) In the case of the basic set H, the topology of pointwise convergence coincides with @H on the set E = RH. I

According to the definition of the projective generation, the projection Pi is always (X Sf;, 8Tj)-continuous. An important supplement to this is the following '

iEI

remark: (7.1.26) If G is Sf-open, then pj(G) is Sfropen for every j E /. Proof. Let x1 Ep/G), i.e. x1 = pj(x), x E G. Then, for suitable Sf;-open sets G;, by (7.1.23) x E X G; c,.G. Evidently x1 E G1 c pj(G). I iEJ

(7.1.27) If A; c E;, A=

X

A;, then the Sf-closure of A is

iE/

X A;, where A; denotes iE/

the Sf;-closure of A;,· JJ A; is Sf;-closed for every i, then A is Sf,;,closed. Proof. The second statement follows from the (Sf, Sf;)-continuity of the projection Pi and the equality A = np;1(A;), Therefore, for arbitrary A; C E1, X .J, .

ITT

~

will be Sf-closed and will contain A (the Sf-closure of A is denoted by A). On the other hand, choosing for x E X A; an arbitrary finite set /' c I and for i EI' IE/

a Sf;-open G; and putting G; = E, whenever i E/ - I', x E X G1 = V implies IE/

that there is a point y 1 E G; n A; for every i; thus y = (y;) EA n V so that x EA,

I

The following theorems follow at once from the general properties of pro.ectively generated topologies. In this way, it follows from (7.1.4): J (7.1.28) Let [X, Sf'] be a topological space and/: X-:"-> £; The map fis (Sf', Sf)continuous iff p 1 of is (Sf', Sf;)-continuous for every i E /. I According to (7 .1.11) : (7.1.29) Let Sf; and

Sf'

=X iEJ

.

sr; be two' topologies on E;, Sf1 < sr; for every . i, Sf = X Sf;, iE/

sr;. Then gr < Sf', I

286

7.1.

(7.1.30)

THE PRODUCT OF TOPOLOGICAL SPACES

(7 .1.30) Let 0 =I= Ai c Ei for all i and A =

X ~I

X (8Ti [ A;). ~I Pi 11; of the pro-

Ai. Then 8T IA =

Proof. The statement foHows from (7.1.7) as the restriction jection P;: E ~ E; is evidently nothing other than the projection of A onto A;.

I

(7.1.31) With the notations of (7.1.30), if A1 = E; for an index j EJ and A; = = {Yi} for other indices, then Pi i A i,r a (8T [ A, 8T1)~homeomorphism. Proof. Let qi = Pi 11; (i EI). The map q1 is evidently bijective. By (7.1.30) 8T I A = X (8Ti ) A;) == sup {q;- 1 (8T; [ A;): i EI}. However, if i =I= j, then q; is iEI

a constant map, thus by (2.6'.16) it is (q1- 1 (8T), 8T; [ A;)-continuous, q;- 1 (8T; / A;) < < q1- 1(8f1). Hence sup {q11(8T; [ A;): i E J} = q1- 1(8T1). Accordingly the statement follows from (2.6.44) (c). I We get from (7.1.12): (7.1.32) A filter base t in E converges to x EE with respect to gr iff, for every i EJ, P;{t) ~ xi = P;(x) with respect to 8T;. I It follows from (7.1.13):

(7.1.33) The product of countably many Mr (M2-) spaces is an Mr (M2-) space as well.

I

·

Let us now add: (7.1.34) The product of countably many separable spaces is separable. Proof. Let Si be a countable 8T;-dense set for every i .E N and choose a point z; EE; for i EN. Let S be the set of those points x EE the coordinates of which are X; = z; with the exception of a finite number of indices i, while for these exceptional indices i, x ES;; in other words S =

LJ

S 1,

JtCJ

where I' runs over the finite subsets of I and

where i EI' implies Ti = S;, i E J - I' implies T; = {z;}. It can be easily seen that all sets Sr and then S too are. countable. On the other hand, Sis ST-dense; for if y EE, y E V = X G;, G; is 8T;-open for i E I' and G; = E; whenever i E J - I' iEI

(I' c I finite), then looking for a point xi E S; n G; for every i E J' and putting x; = z; whenever i E J - I', x = (x;) E Sr c S and x E V. I It can be seen from (7.1.14) that:

(7.1.35) The product of S1-, S 2-, Sa-, or S"-spaces has the same property. Let us add:

I

(7.1.42)

7.1.d.

287

THE PRODUCT OF COMPACT SPACES

(7.1.36) The product of T 0 -spaces is a T 0 -space as well. Proof. If x, y EE, x -:/= y, then X; -:/= Yi at least for one i E /. Let e.g. G; be a 8f;-open 8et such that X; E G;, Y; ~ G;. Then G = p 11(G;) is Sf-open, x E G, y~ G. I

It follows from (7.1.35) and (7.1.36): (7.1.37) The product of T1-, T 2-, T 3 -, Tn-spaces has the same property.

I

The following theorems deal with the mappings of product spaces. (7.1.38) Besides the usual notations, let [E/, sr;J be another topological space for every i, J;: E; ~ E/ (Sf;, sr;)-continuous, E' = XE;, Sf' = X sr;, p;: E' ~ E{ iE[

iE[

the ith projection and f: E ~ E' the map for which J; o P; = p; of f is then (Sf, §"')continuous. Proof. By (7.1.28) it is enough to prove the (Sf, sr;)-continuity of p; of for every i E /. However this follows from the (8T, 81";)-continuity of Pi and the (81";, 81";)continuity of J;. I

(7.1.39) If, under the hypotheses of (7.1.38), J; is a (Sf;, 8T!)-homeomorphism for every i, then f is a (Sf, §"')-homeomorphism. Proof. If g:E' ~Eis defined by the formula J;- 1 op{ =P;og, then gof is the identity map of E, fog that of E', so that f is bijective and g = J- 1. The (8T, §"')-continuity of/and the (8T', 81")-continuity of g result at once from (7.1.38). I 8T

(7.1.40) With the notations of (7.1.19), let 8T; be a topology on B;, 8Tf(i) = Sf;, = X 81";, 8T' = X 81"1. Then g is a (8T, §"')-homeomorphism. iEJ

jEJ

Proof. The /(i)th coordinate of g(b) is equal to b; which is obtained from b by the projection from B onto B; i.e. in a (8T, 81";)- or (8T, 8T/(i))-continuous way. Hence g is (8T, 8f')-continuous and it can be similarly seen that g- 1 is (8T', 81")continuous. I (7.1.41) With the notations of (7.1.20), let 8T; be a topology on A;, 8T

X 81";, 1= iE/;

8T

8T'

= X jEJ

1. Then f

81"

= X

81";,

iE/

is a (8T, §"')-homeomorphism.

Proof. By applying (7.1.28) repeatedly, from the (8T, 81";)-continuity of P; (i E /) the (8T, sr;)-continuity of qj of and from this the (Sf, §"')-continuity of f follows. On the other hand, by rR o {j_j = P; o1- 1 , from the (8T', 8f;)-continuity of qj and the (8Tl, 8f;)-continuity of rji the (8T', 8f;)-continuity of the left-hand side results which implies the (8T', 81")-continuity of 1- 1 . I 7.1.d. The product of compact spaces. A series of important theorems deals with the question of how the different generalizations of compactness behave relative to the construction of product spaces. (7.1.42) Tykhonov's theorem. The product of compact spaces is compact.

288

7.1.

(7.1.43)

THE PRODUCT OF TOPOLOGICAL SPACES

Proof. We refer to (5.3.1). If u is an ultrafilter in E, then by (6.1.42) p;(u) is an ultrafilter in E; and hence p;(U) -+ X; EE; with respect to Sf;. With the notation x = (x;), u-+ x with respect to Sf by (7.1.32). I

From this we get at once by (7.1.22): (7.1.43) The product of a finite number of locally compact spaces is locally compact. m

Proof. If x E X E; and m

(7.1.22)

X Vi

Vi

is a 8i"rcompact neighbourhood of

1

is a Sf-neighbourhood of x, further

1

X;,

then by

m

X

(Sf; J V.) is compact by

1

(7.1.42). By (7.1.30) the latter topology is identical with (Sf

m

I X V;). I 1

(7.1.44) The product of countably many sequentially compact spaces is sequentially compact. CJ)

Proof. Let (xn) be an arbitrary sequence in the set

X E; = E.

This has a sub-

1

sequence ( 1 = (x1n) such that Pi(X1n) -+ Yi with respect to 8J1. The sequence ( 1 = (x 1 n) has a subsequence ( 2 = (x2 n) such that plx2 n) -+ Y2 with respect to Sf2 • In this way we get sequences (k = { Xkn: n EN} such that (k is a subsequence of (k-l and pixkn)-+ Yk EEk with respect to Sfk whenever n-+ oo. Consider the sequence ( = (xnn). It is a subsequence of (xn) and beyond the kth member it is also a subsequence of (k, therefore, for every k, Pk(xnn) -+ Yk with respect to Sfk if n -+ oo. By (7.1.32) Xnn -+ y = (y;) EE with respect to Sf. I In order to use it later, it is worth-wile to formulate the essential thought of the preceding proof called "the principle of diagonal selection". (7.1.45) Let (o = (xn) be an arbitrary sequence, ( 1 , ( 2 , ••. subsequences of (o such that (k = { xkn : n EN} is a subsequence of (k-l (k = 2, 3, ... ). The sequence ( = (xnn) is then from its kth member on, a subsequence of (k for every

kEN,

I

(7.1.46) The product of a countably compact and a sequentially compact space is countably compact. Proof. Let [X, 8f1] be sequentially compact, [Y, 8J2] countably compact, and {(xm Yn) : n EN} c Xx Yan arbitrary sequence. The sequence (xn) has a convergent subsequence, say Xn;-+ x EX. Let y be a cluster point of the sequence (YnJ If V is an arbitrary (8f1 x 8J2)-neighbourhood of the point (x, y), then x has a 8J1-neighbourhood V1 and ya 8J2-neighbourhood V2 such that V1 x V2 c V. For a suitable i0 , i ~ i0 implies Xn; E V1 and Yn; E V2 for infinitely many i, so that (xn;, Yn) E V for infinitely many i. I (7.1.47) The product of real-compact spaces is real-compact. Proof. Let 'P; be the family of 8lrcontinuous functions, '1i that of 8f-continuous

7.1.d.

(7.1.47)

289

THE PRODUCT OF COMPACT SPACES

functions. If r is a 6U'.g;-Cauchy filter base in E, then P;(t) is a 6U'.g;rCauchy filter base inE; for every i since, for any function fE i. Hence t E [a;, b;) for every i, in particular t E H and the sequence (xn;(t)) is identical with the non-convergent sequence (I, 0, 1, 0, ... ). On the other hand, let (Yn) be an arbitrary sequence in A, Mn C H the countoo

able set outside of which Yn = 0, and M =

U Mn1

19 Akos Csaszar

Let us order the countable set

290

7.1.

THE PRODUCT OF TOPOLOGICAL SPACES

(7.1.48)

M into sequence M = {t;: i EN}. Let ~1 = (Yin) be a subsequence of the sequence ~o = (Yn) such that n -+ oo implies Yin(t 1) -+ a1 where O ~ a 1 ~ 1; there exists such a subsequence by O ~ Yn(t 1 ) ·~ I according to the BolzanoWeierstrass theorem. Now let ~2 be a subsequence of ~1 which converges at t2 to a limit O ~. a 2 ~ 1, and in general ~; a subsequence of the sequence ~i-l which converges at t;. The sequence ~ = (zn) constructed according to (7.1.45) is a subsequence of ~0 and a subsequence of ~; from the ith member on and converges therefore to a; at t;. As zn(t) = 0 for t EH - Mand for every n, (zn) converges pointwise to that function z EA for which z(t;) = a; and z(t) = b whenever tEH- M. I

7 .1.e. Embedding theorems. The definition of the product topology was introduced as a special case of the projectively generated topology. It is an important fact that, with some restrictions, the projective generation can be reduced to the construction of the product topology since the projectively generated topology is homeomorphic with a restriction of the product topology, i.e. the projectively generated space can be topologically embedded into the product space. The following theorem deals with this fact:

(7.1.48) For I # 0, iE I, let [E;, 8!;] be a topological space, X # 0 a given set, J;: X-+ E;, §"* the topology projectively generated by the system {Ji, W;: i EI}, E = X E;, §" = X 8!;, P;: E -+ E; the ith projection, and f: X -+ E the mapping iEl

iEl

for which the ith coordinate off(x) is fi(x) EE; whenever x EX, finally h = f I fjX). If we suppose that there exists, for x, y E X, x # y, an i EJ such that fi(x) # fi(y), then h is a (w*, §" I f(X))-homeomorphism. This condition is fulfilled if§"* is a T 0 topology. Proof. Pi of= Ji implies §"* = f- 1(§") by (7.1.6) and it follows then from (2.6.44) (b) that h is a homeomorphism since f is injective by hypothesis and h

is bijective. If §"* is a T0-topology, then there exists, for x =I= y, x, y EX, e.g. a §"*-open G such that x EG, y ~ G. Then there are by (7.1.3) a finite number of indices ij El n

and W;ropen sets G;; such that x E

n fi;1(G;;) c

G. Thus y ~fi; 1(Gt;) at least

j=l

for one j, i.e. fi;(x) EG;;, fi;(y) ~ G;;-

I

It follows at once from (7.1.48):

(7.1.49) Let [X, §"*] be a Tykhonov space; ak, _ 2 > ... from points belonging to Min such a way that the distance of consecutive members is smaller than

_!_ and lim 2

akm = ak. All this can be

m--oo

done since Mis dense. Finally we get points akm EM such that O < ak,m+I - akm
0 less than the ex,y occurring here, then U U,, ... Jn},, c U.] 17. Let [E, Sf] be a compact topological space and 'P a function family which consists of 8f continuous functions and contains all constants; if x, y EE, x # y, then there exists an /E

& = X &i, &1 = X &i, &' = X &1. Then f is a (&, &')-equimorphism, I IEI iEI1

iEJ

.

.

It follows from (7.2.12) that: (1.2.22)

If & = X &,, then 8', = X sr,,j' I iEI

iEI

From this and from (7 .1.36) follows: (7 .2.23) The product of separated proximity spaces is separated as well.

I

7 .2.c. Embedding theorems. The following theorem corresponding to (7 .1.48) is valid: (7.2.24) Let I =fa 0, [E1, &d be a proximity space for i EI, X =ft 0 a given set, fi: X--)- E1, &* the proximity projectively generated by the system {Ji, &1: i EI}, E ~ X E;, & = :X Si'i, p 1 : E --)- E1 the ith projection andf: X --)- E the map for which iEJ

iEI

p 1 of= Ji, finally h = f I §X). If x, y E X, x =fa y implies fi(x) =fa fi(y) for at least one i E1, then h is a ( &*, Si' If(X) )-equimorphism. This condition is certainly fulfilled if &* is separated. Proof. We have from (7.2.6) that &* = 1-1( ~ and (3.1.50) (b) shows then that h is an equimorphism since it is bijective. If &* is separated and x, y EX, x :f= y, then {x} &* {y} implies by (7 .2.3) that there is an i E1 for ·which {Ji(x)} &1 {fi(y)}, hence fi(x) =ft fi(y), I It follows from (7.2.24) by (4.2.23) that: (7.2.25) Let [X, &*] be a separated proximity space, I = [O, l], & = @Pt I I, p1 the Euclidean metric on R. Then, for suitable I, [X, &*] is equimorphic with a subspace of the space [11, &1 ]. · Proof. If {g1: i El} is the family of all proximally continuous functions whose values are in I, then, by (4.2.23) and (4.2.15), &* is the proximity projectively generated by the system {g1, &Pt: i EI} and, according to (3.1.47), the same can

be asserted with the notation Ji = g1 11 for the system {Ji,&: i EI}.

I

Of course, by (7.2.22), &1 occurring here is nothing other than the (unique) proximity of the Tykhonov cube [11, (~ I 1)1]. The space [11, &1 ] is compact by (7 .1.42), hence the closure of one of its subspaces furnishes the compactification of the subspace: in question. This - with restriction to separated proximity spaces - gives another proof of the existence of the Smirnov compactification. 7.2.d. Exercises. 1. With the notations of exercise 7.Lf.1, let @1 be a proximity on E1, p 11 (&i, &,)-proximally continuous and E =fa 0; Then the proximity & projectively generated by the system .{Pt, &1 : i EI} is. called the projective limit of the system {&1,p1J. Show that 8', coincides with the projective limit of the

(7.2.25)

7.2.c.

system {8!&;,Pu} and§'=

EMBEDDING THEOREMS

X §'t [ E,

305

further prove the statements corresponding

iEl

to the statements (b) and (c) of the exercise m~mtioned above. 2. Let §'* be the proximity projectively generated by the system {Ji, W';: i EI} and W'; separated for every i. Show that §'* is separated iff, for x i= y, there is an i EI such that f;(x) i= fi(y). 3. Let §' be the Cech-Stone proximity on R belonging to the topology $, further · A= {(x,y):x + y ~ O}, B = {(x,y):x + y~ 1}. Show that (a) A &2 B; (b) A and B are far from each other with respect to the Cech-Stone proximity of $ 2 • [The reasoning of p. 169 can be applied.] 4. Show that, for the sets A and B of the preceding exercise, A §' 2 B holds even if §' denotes the discrete proximity of R. Accordingly the product of two discrete proximity spaces need not be discrete although the product of two discrete topologies is discrete. 5. Let§' be the coarsest proximity on R inducing$ and§'' the coarsest proximity on R 2 inducing @2 • Show that the projection of R2 on any of its factors is not (W'', &)-proximally continuous, hence §'' i= §'2 • [If A= {(x,O):x~ 1}, B = {(O,y):y~ 1}, then AW'' B, but Pi(A)@pi(B). P2(A) §' P2(B).] 6. Let §'* be the proximity projectively generated by the system {/;, Sf;: i EI}. Show that the filter base r is &*-compressed iff f;(r) is &;-compressed for every i. m

[If A§'* B, then A =

UAj, B

j=l

n

=

UBk, so that,

for every pair j, k, fi(Aj) W';U,k)

k=l

fi(Bk). Therefore there exists an Rjk Er for which f;(Rjk) intersects at most one of fi(Aj) and /;(Bk), thus Rjk can intersect at most one of A1 and Bk. If R E r is m

such that R c

n

n nRjk, then R can intersect at most one of A and B.] j=l k=l

7. Let [E, §'] be a proximity space. Show that A §' B iff A x B (§' x §') D where D = {(x, x): x EE} c Ex E. [Let [E', §''] be the compactification of [E, &], 8f = Wi;,, W' ~ 3li;,,. Then if A§' B, there is p EA n JJ by (6.4.18) where A is the W'-closure of A. By (7.1.27) (p, p) EA x JJ = A x B (by this the (SJ' x 31'')-closure of A x Bis understood). Clearly (p, p) E fj so that fj (§'' x @') A x B by (P 2) and (7.2.22) and D (@ x SY) A x B by (3.1.14) and (7.2.16). On the other hand, if A§' B, then by (P 6) there are M, N such that M u N = E, M §' B, N §'A; then D = M' u N' where M' = = {(x,x):xEM}, N' = {(x,x):xEN} and p 2(M')&p 2(A x B), Pi(N')& Pi(A x B) so that D & x SY A x B by (7.2.13).]

20 .Akos Csaszar

306

7.3 ..THE

PRODUCT OF UNIFORM SPACES

(7.3.1.)

7.3. THE PRODUCT OF UNIFORM SPACES 7 .3.a. Projective generation of uniformities. Definitions and theorems can be established in a way analogous to the generation of topologies and proximities. By (3.2.27) and (3.2.39): (7.3.1) Let I# 0, E # 0, [E;, 611;] be a uniform space for every i EJ, and f;: E---+ E;. Then there exists a coarsest uniformity 611* onEfor which every Ji is (611*, 611t)uniformly continuous, namely (7.3.2) 611* in (7.3.2) is called the uniformity projectively generated by the system

{Ji, 611;: i E J}. In the following we shall always use the notations of (7.3.1). E.g., by (4.2.12), the uniformity 611 induced by the function familycp - with the notation

0, consider the surrounding U E "lf 2 for which ((x, y),

(u, v)) E U holds iff (x, u) E Ua e, (y, v) E Ua e, i.e. iff a(x, u) < ~, a(y, v) < ~'2~ 'z 2 2 Then ((x, y), (u, v)) E Uimplies a(x, y) ~ a(x, u) + a(u, v) + a(v, y) < a(u, v) + 8, a(u, v)

~

a(u, x)

thus I a(x, y) - a(u, v) I
(e): Let 21: = {An: n EN} be an open cover and 58 /4; 21: a closed, conservative cover. If

en=

U {BE58:BcA,.},

then (£ = {en: n EN} is a closed cover by (8.3.42) and Cn c An. (e) => (f): This is clear from de Morgan's identities (1.1.5). (b) => (h): Let [E', 8T'] be a compact pseudo-metrizable space. By (5.2.21), (3.2.68), and (2.4.16), there is a countable base S) = {Hn: n EN} for Sf'. It can be assumed that Hi, Hi ES) implies Hi u ¼ES) by replacing, if necessary, the elements of S) by their finite unions. For Sc E x E', x EE, let S(x) denote the set of those y EE' for which (x, y) ~ ES. Now consider two disjoint closed sets P and Q in E x E'. Let us define, for n EN, For x EE, P(x) and Q(x) are clearly Sf'-closed and disjoint and, for y EP(x), there is a Hi E S) such that y EHi c Hi c E' - Q(x) (Sf' is regular by (2.5.40)). Since P(x) is 8!'-compact by (5.3.4), a finite number of these Hi cover P(x) and if Hn E S) is their union, then x E A,.. Hence 21: = {An : n EN} is a cover of E. On the other hand, if x EAm y EE', let Uy and Vy be open neighbourhoods of x and y respectively such that ·

= 0 if y EP(x), Vy) n P = 0 if y EQ(x),

Vy c Hn and (Uy x Vy) n Q Vy c E' -

ll,. and (Uy

(Uy x Vy) n (Pu Q)

x

= 0 if y

~ P(x) u Q(x).

(8.3.56)

8.3.i. k

If E' =

LJ Vy1 and

COUNTABLY PARACOMPACT SPACES

363

k

U =

1

n Uy,,

then U is an open neighbourhood of x and, for

1

z E U, y EE', y E P(z) implies y E Vy, for some i, hence (UYI x V:,) n Pi= 0, Vy,~ Hm_JJ E Hn, y E Q(z) implies y E VYI' (Uy, x Vy) n Qi= 0,Vy, c E' - llm

yEE - Hn. Therefore Uc An and An is open. By (b), let )S = {Bn: n EN} be a locally finite open cover of E such that Bn c c An. Let us define

Then M is (gr x gr')-open. If (x, y) E P, then x E Bn for some n, hence x E An, y EP(x) c Hm (x, y) E Bn x Hn. Therefore P c M. It suffices ·to show that Q C (E x E') - M. Suppose (x, y) EQ n M. There is a neighbourhood W of x which intersects only a finite number of the sets Bm say Bn , ... , Bnk· The (gr x gr,)-neighbourhood W x E' cannot intersect the sets Bn x Hn except those Bn, x Hn,, i = 1, ... , k. Hence (x,y) E Bn, x iln 1 for some i by (7.1.27). However, ~ E Bn, C An, and y E Q(x) contradict y E lln 1, so y ~ Q(x) which was to be proved. (h) =>. (i): Obvious. (i) => (f): Let I = [O, 1], gr'=~ I I and let gr x gr, be normal. By (7.1.31), [E, gr] is homeomorphic with [E x {O}, gr' IE x {O}] so that gr is normal by (2.5.32). 00

On the other hand, suppose that F 1 gr_c10sed. Then

::)

F2

::) ••• ,

nFn =

0, and each Fn is

1

Hn = (E - Fn) x [ 0, : ) is (gr x gr')-open, 00

P

= (E x

I) -

LJ Hn 1

and Q =Ex {0}

are (gr x gr')-closed, and P n Q = 0 since x EE implies x EE - Fn for some n and then (x, 0) E Hn- By the normality of gr x gr,, there are (gr x gr')-open sets M and N such that Pc M,

Q c N,

Mn N = 0.

Let us define .

00

Then Gn is clearly gr_open and x oo

with (x, 0) EQ. Hence

En Gn would imply

nGn = 0. 1

1

(x,

0) EM in contradiction

364

8.3.

(8.3.57)

PARACOMPACT SPACES

It suffices to show that Fn C Gn for each n. Now if x E F,,, then x

for i~n,:

Efro,+)

(x, :)Et

for i>n, hence

Ef E

- F;

ys;,(x, :)EPcM,xEGn

as stated. (f)

n

= (g): Let m= {An: n EN} be an open cover, Fn = E -

U Aj. Then the 1

00

sets Fn are closed, Fn:) Fn+l and 00

and

n1 Gn = 0.

n1 Fn = 0. Let Gn be open such that F,, c

Select 8T-continuous functions fn: E--+ R such that O ~ fn(x)

Gn ~

1

for x EE, fn(x) = 0 for x E Fn and fn(x) = 1 for x EE - G,,. Let us finally define 1 f = ~ 2n fn; then O < f(x) ~ 1 for x EE and f is 8T-continuous by (2.6.30) and 00

(2.6.28). Now the sets

Bn; = {x:fnCx) > 0, ~; 0

of E. In fact, x EE implies ~; 0, hence x Ef Fn and x E Aj for some j ~ n. Moreover, the star-finite property of the sets B,,; clearly implies the same property for the sets cnij• (g) (a): This is an obvious consequence of (8.3.55).1

=

It is worth while to mention that the construction of a normal space which is not countably paracompact is very difficult. A remarkable consequence of (8.3.56) is the fact that the countable variant of paracompactness coincides, for normal spaces, with the countable variant of metacompactness (cf. (a) and (d)). Moreover, the same can be said for the condition of the existence of a star-finite refinement which is an open cover: for an arbitrary countable open cover of a normal space, this condition is (by (g)) equivalent to countable paracompactness. It is natural to ask what can be asserted when the same condition is postulated for arbitrary (not necessarily countable) open covers. This will be the subject of the next section.

(8.3.59)

8.3.j.

STRONGLY PARACOMPACT SPACES

365

8.3.j. Strongly paracompact spaces. A topological space (or a topology) is said to be strongly paracompact if, for every open cover, there exists a star-finite refinement which is an open cover as well. The terminology is motivated by:

(8.3.57) A strongly paracompact space is paracompact. Proof. (8.3.55). I

In order to formulate an equivalent characterization, let us call a system of sets mstar-countable if each A E mintersects at most countably many elements of m. A star-finite system is clearly star-countable. Now the following theorem holds: (8.3.58) For a topological space [E, 8T], the following statements are equivalent: (a) 8J is a strongly paracompact S2 -topology; (b) 8J is regular and, for each open cover m, there is a star-countable refinement 58 /4; mwhich is an open cover as well. Proof. (a)=> (b): (8.3.57) and (8.3.15). (b) => (a): Let m be an arbitrary open cover and 58 ~ m a star-countable open cover. For the elements B E 58, let us define an equivalence relation as follows: let B'"" B' hold if there exists a finite number of sets B; E 58 (i = I, ... , n) such that B 1 = B, Bn = B', and B; n B;+ 1 i= 0

(8.3.59)

= I, ... , n - l. This is clearly an equivalence relation. By (1.1.7), 58 is the union of disjoint equivalence classes 58s (s ES). Consider the sets

fnr i

They are open and disjoint since BE 58., B' E 581, B n B' i= 0 clearly implies = t. On the other hand,

B '"" B' and s

sES

so that each set Es is closed. Fors ES, select a set BE 58s and denote by 58sm for n EN, the system of those B' E 58s for which there are B; E 58 (i = I, ... , n) satisfying (8.3.59). By means of induction with respect to n, it is easy to see that 58sn is countable for n EN 00

and as 58s =

U 58sm 58s

is countable as well. Denote

n=l

58s = {Bs;: i EN}, C£; = {Bs;:sES}. Then

8.3.

366

(8.3.60)

PARACOMPACT SPACES

and Bsi c E. implies that ~i is discrete. Hence gr- is paracompact by (8.3.15), and gr- IE. is paracompact as well by (8.3.21); A fortiori, SJ' IE. is countably paracompact and is normal by (8.3.15) and (8.2.15) so that, by (8.3.56), there is a star-finite gr- I E.-open cover (i): Let 58 be a point-regular base; it can be supposed that the sets from 58 are not empty. Denote by 'iJc the system of sets minimal in 58 (i.e. of those sets NE 58 for which B c N, B E 58 imply B = N); of course 'iJc = 0 can hold as well. Let now ?S1 = )S, W11 be the system of sets maximal in ?Si, and if Wei, ... , Wcn-l c

m;

m

n-1

c ?S are already selected, let ?En = (?E - U Wei) u 'iJc and Wen be the system of sets 1

maximal in ?Sn- It will be shown that Wen is a cover of E for every n. For this purpose we have only to see, by (8.4.2), that every ?S,, is a cover of E. However it can be seen from (8.4.2) that every We; is point-finite, thus if a point x EE belongs to infinitely many sets from

m,

n-1

then there is even in

m- u We;

a set contain-

1

ing x. On the other hand if x belongs only to a finite number of sets from ?S, then their intersection N is the smallest neighbourhood of x and belongs itself to as is a base. N is then minimal in i.e. X ENE 'Efc, since if B E QJ, BC N, B # N would be true, then necessarily x ~ B would hold which is impossible since, by choosing a point y E B, y would have a neighbourhood not containing x and on the other hand every neighbourhood of x would contain y. We shall see that, if x EE, then {Wcn(x): n EN} is a neighbourhood base of x. Let V be an arbitrary neighbourhood of x. If x ENE 'iJc and BE ?S is such that x EB c V n N, then B = N, thus N c V. Suppose now that WcnCx) c V does not hold for any n. Then, for every n, there is a set Mn E Wen such that x E Mm Mn n n (E - V) # 0. Since in the preceding Mn ~ 'iJc and. n # m clearly implies Wen n n Wcm C 'Efc, therefore in this case Mn # Mm· However this is impossible since only a finite number of sets from )S and containing x can intersect E - V. By (8.3.39), for the point-finite open cover Wen of Ewe can construct a locally finite open cover SU:n such that SU:n ~ Wcw Then SU:n(x) c Wcn(x) for every x EE and hence {2:fn(x): n EN} is also a neighbourhood base of x. From this it follows

m m

m,

00

at once that if, for an open cover Cf of E, we look for the sets in the system

U mn 1

which are contained in CE Cf, then we obtain a er-locally finite open cover of E

376

8.4.

METRIZATION THEOREMS

(8.4.6)

which is a refinement of (£. As by (8.1.5) sJ" is an S4- and a fortiori an S3-topology, therefore by (8.3.15) it is fully normal. (j): This is evjdent by (8.2.15). (i) (j) (a): Let (58n) be a development, U1 = UfS and if the sI"2-open surrounding Un c E x Eis already chosen, let Un+l be a sJ" 2-open cover for which Un+l o o Un+i c Un n U'i8n+i· Clearly {Un: n EN} is a uniform base which generates a pseudo-metrizable uniformity 6U by (4.2.33). As by (7.3.47) x EE, n EN implies that Un(x) is sJ"-open, therefore sJ"'ll < 8'. On the other hand, by Un(x) c U'i8.(x) = = 5Bn(x), sJ" < sJ"'ll holds and sJ" = 8f'Ll. (i) (k): Let again (58n) be a development, fil1 = 58i, and if the open cover mn is already chosen, let mn+l be an open cover such that stfiln+l ~ 91n (n) 58n+l· Then by (8.2.11) (b) mn ~ 58m thus filn(x) c 5Bn(x) for every n, so that (filn) is a development as well; further by (8.2.12) filn+l ~ filn. (k) (a): If 58n+I ~ 58n in the development (58n), then by (8.2.6), with the notation Un = U'i8., {Un: n E N} will again be a uniform base and, for the pseudometrizable uniformity 6U generated by it, sJ"'Ll = 8'. (a) (h): Let, in the pseudo-metric space [E, p ], 58n be the system of balls having the form Up, 2 -n(x). If Vis an arbitrary neighbourhood of x and Uµ, 2 -n(x)c c V, then by defining V' = Up, 2 -n-,(x) evidently 5Bn+lV') c Uµ, 2 -n(x) C V. (h) (i): Let (58n) be a sequence of covers having the property given in the hypothesis and filn = 581 (n) ... (n) 58n. Then filn is an open cover as well, filn+l ~ filn for every n, and, with filn ~ 58n and because filn(V') c 5Bn(V'), (filn) has the same property as (58n). We will show that sJ" is fully normal. Let(£ be an arbitrary open cover of E, ':.D the system of those open sets D for which there exist n E N, A E film and CE (£ such · that D c A and 5len(D) c C. To each one of these D select and fix a number n of the above property. ':.Dis an open cover of E since, for x EE, there is a CE(£ such that x E C and then an open neighbourhood V of x and an n E N such that filnCV) c C; if x EA E mm then, with the notation D = A n V, x ED E SD. On the other hand, if x EE and n0 is the smallest ayiong the numbers n assigned to the sets D from ':.D containing x and n0 belongs to the set D0 E ~ so that x E D0 , then

= =

1,

=

=

=

=

00

':.D(x)

C

n filn(x) C

filn,(X) C filn,(Do)

n=n 0

and the latter set is contained in a CE (£. Accordingly st ~ ~ (£, We show finally that (91n) is a development. Indeed if Vis an arbitrary neighbourhood of x and another neighbourhood V' of .x and n E N are chosen in such a way that filn(V') c V, then a fortiori filn(x) c V. I (2.5.5) shows that by adding axiom (T0) to an arbitrary condition of (8.4.5) we get a necessary and sufficient condition for the metrizability of the space. 8.4.d. Applications. It follows at once from the condition (8.4.5) (b) as a generalization of the statement (7.3.36) that:

(8.4.6) Every regular M 2-space is pseudo~metrizable. I

(8.4.11)

8.4.d.

377

APPLICATIONS

On the other hand, by (8.4.5) (e) and (8.3.9), we can say that: (8.4.7) Every pseudo-metrizable Lindelof space is an M 2-space.

I

According to (8.3.23) we can say that: (8.4.8) If, in the topological space [E, 81"], E = U E;, every E; is open, E; n Ej = iEI

= 0 if i ¥= j, and 81" I E; is pseudo-metrizable for every i, then

81" is pseudo-metrizable as well. Proof; Taking, for every i, a a-locally finite base 5.8; of 8f I E; evidently m= = U 5.8; will be a a-locally finite base for 8f and the statement follows from iEI

(8.4.5).

I

(8.4.9) A locally compact S 2-space [E, 81"] is pseudo-metrizable iff E = U E;, .

iEl

E; is open, i ¥= j implies E; n Ei = 0 and 8f I E; is an M 2-topology for every i. Proof. If 8f is pseudo-metrizable, then it is paracompact by (8.3.16) and hence by (8.3.24) E can be written as the union of disjoint open sets E; in such a way that 8f I E; is a-compact and hence Lindelof by (8.3.20) for every i. As of course 8f IE; is pseudo-metrizable, it is therefore an M 2-topology by (8.4.7). Conversely, if Eis of the type formulated in the theorem, then 8f IE; is regular (81" being regular by (5.3.54)) and hence it is pseudo-metrizable by (8.4.6); then by (8.4.8) 8f itself has the same property. I 8.4.e. Metrizability of proximity spaces. It is worth-while to notice that a condition analogous to condition (8.4.5) (k) can be given for the (pseudo-)metrizability of proximity spaces, i.e. for the validity of & = SfP for a suitable pseudo-metric (metric) p. For this purpose, let [E, &] be a proximity space; the cover ~ of E will be called a @-cover if, for any set A c E, A iE - ~(A).

(8.4.10) The proximity space [E, Sf] is pseudo-metrizable iff a sequence (~n) of &-covers of E can be given for which ~n+l ~ ~n and A &B implies ~nCA) n B = 0 for suitable n. Proof. If & = &P for a pseudo-metric p and ~n is the cover consisting of the balls S(x, 2-n), then the conditions of the theorem are evidently fulfilled since p(A, E - ~nCA)) ~ 2-n as soon as A ¥= 0, E - ~nCA) ¥= 0, ~n+ 1(x) C S(x, 2-n), and 2-n < p(A, B) implies ~n+iCA) n B = 0. On the other hand, let now (~n) be a sequence of covers with the given property and Un = Urs,n c E x E. By (8.2.6) Un+i o Un+i c Un so that {Un: n EN} is a uniform base inducing a uniformity 611, pseudo-metrizable by (4.2.33). Since by (8.2.7) Un(A) = ~nCA) for every set Ac E, therefore A i B holds iff Un(A) n n B = 0 for at least one n so that &'Lt = &. I By (2.5.5) and (3.1.16) we can add: (8.4.11) The proximity space [E, &] is metrizable iff it is separated and there is a sequence of covers with the property formulated in (8.4.10). I

8.4.

378

(8.4.12)

METRIZATION THEOREMS

8.4.f. Continuous closed images of metrizable spaces, As a further application of the metrizability conditions we shall examine the question under what conditions the image of a metrizable space will also be metrizable. The metrizability of discrete spaces shows that only the continuity of the map is not in general sufficient for this. According to the following theorem, even in the case of a · continuous closed map some further restrictions ·are needed: (8.4.12) Let [X, 8T1 ] and [Y, 8f2 ] be topological spaces, 8T1 (pseudo-Jmetrizable, X ~ Ya (8Ti, Sf2)-continuous and closed map. The following statements are equivalent: (a) 8f2 is (pseudo-)metrizable; (b) 8f2 is an M 1-topology; (c) If y E Y, then mar 1- 1(f) is 8Trcompact. If8T1 is metrizable, then we can write in (c)marf- 1(y) instead ofmarJ- 1(y). Proof. (a)=> (b): Obvious. (b) => (c): If H = marJ- 1(y) cf- 1(y) were·not8f1-compact, then neither could it be countably compact by (5.3.33) and hence by (5.3.31) there would be a sequence (xn) in H which has no cluster point. Let p be a pseudo-metric on X such that 8f1 = Sfp, and (by using (2.4.13)) let (Vn) be an open 8f2-neighbourhood base of y such that Vn+l c Vn (n EN). By (7.4.14) 8f2 is an S1-topology and hence y c Vn for every n, thus 1- 1(Vn) :J H. Let us select points Zn such that

f:

Z;, Ef- 1(Vn)-J- 1(f),

p(Xn,

·.

1 Zn) < -.

n

00

Then evidently (zn) has no cluster points either, hence F = U Zn is a closed set 1

for, if x EX - F, then a suitable open neighbourhood of x can contain only a finite number of Zn and at the same time it ~an only intersect a finite number of zm and discarding these zn from it we obtain a neighbourhood of x not intersecting F. Since Zn EX - 1- 1(f), and gr-1 is an Si-topology, zn C X -f- 1(y), thu, /(zn) n n Ji = 0 for every n and f(F) n y = 0. On the other hand, by f(zn) E Vn it follows that f(zn) ~ y, which is impossible asf(zn) Ef(F) and/(F) is closed, y ~f(F). (c) =>(a): Let again p be a pseudo-metric inducing 8f1 and, for y E Y, h EN, Z(y)

= f- 1(y),

H(y)

= mar

Z(y) c Z(y),

Gn(y) = Z(y) u {x: p(x, H(y)) < 2-n}.

If H(y) = 0, then the second member here is to be replaced by the empty set, thus in this case Gn(y) = Z(y) for every n. Gn(y) is clearly 8f1-open and hence, with the notation Vn(y) = Y - f(X - Gn(y)),

we have

(8.4.12)

8.4.f.

379

CONTINUOUS CLOSED IMAGES OF METRIZABLE SPACES

and with Vu(y) open it is a 8f1-neighbourhood of y. Moreover {Vu(y): n EN} is a neighbourhood base of y for if V ¥- Y is an arbitrary open neighbourhood of y, then with 3l2 an Si-topology, y c V, H(y) c Z(y) cJ-1(V), thus H(y) = 0 implies Gn(y) = Z(y) cJ- 1(V) for every n, while if H(y) ¥- 0, then with H(y) compact there is an e > 0 such that x E H(y) implies p(x, X - 1-1(V)) ~ e, thus 2-n < e implies Gu(y) cJ- 1(V) and in both cases Vu(y) cf(Gn(y)) c V. It is clear also that Gn+I (y) C Gu(y).

Let >Sn= {Vn(y): y E Y}.

This is an open cover of Y. We show that the covers >Sn constitute a development. For this purpose, let y E Y, m EN. We have to show that >Sn(y) c Vm(Y) for a suitable n, i.e. that there is an n EN such that u E Y, y E Vn(u) imply Vu(u) c C Vm(y). NowfromyE Vu{u)itfollowsthaty c Vn(u), Z(y) c Gn(u), but on the other hand, Gn(u) c Gm(Y) implies Vn(u) c Vm(y). Thus we have to see that Z(y) c Gn(u) implies Gn(u) ~ Gm(Y) for suitable n. Suppose on the contrary that, for every n ~ m, there is a un E Y such that Z(y) C Gn(un), but Gn (un) - Gm(Y) ¥- 0. Then Y n Un = 0 as y n Un ¥- 0 would imply by (SJ y = um i.e. Z(y) = Z(uJ, GnCunJ = Gn(y) which is impossible if n ~ m. Hence Z(y) n Z(un) = 0 holds as well. If H(un) = 0 would hold, then GnCun) = Z(un), thus Z(y) C Z(uJ would follow in contradiction to the preceding. Hence H(un) ¥- 0 and by Z(y) n Z(un) = 0 Z(y) C {x: p(x,H(un)) < 2-n}.

Let x E Z(y) and Xn E H(un) be such that p(x, Xn) < 2-n. Then Xn -t x, thus f(xn) ~ f(x) and, in view of f(xn) E um f(x) E y, every open neighbourhood V of y contains f(x) and together with it f(xn) and Un too for n large enough (we recall here again (SJ which implies f(xn) = un). If there is now an n > m such that Z(un) c Gm+ 1(y), then x E H(un) implies p(x, H(y)) < 2-m- 1 and from here GnCun) c Gm+lY) u {x: p(x,H(y)) < 2-m-t

+ 2-n} C

Gm+lY) u Gm(Y) = Gm(y).

Accordingly there is, for every n > m, an Xn E Z(un) such that Xn then Xn n Gm+1CY) = 0.

~

Gm+lY) and

00

If this sequence (xn) has no duster point, then F=

U Xn is a closed set and hence m+l

00

f(F) is closed as well; further f(F) c ·

U Um thus y nf(F) = 0 and according to

the

m+l

preceding Y - f(F) would contain un for n large enough; a contradiction. If (xn) has a cluster point x, then it also has a subsequence (xnk) converging to x and .of course x

f Gm+ 1(y).

00

N6w F =

x u U Xnk 1

will be a closed set for if

380

z

8.4.

(8.4.13)

METRIZATION THEOREMS

EX - F, then by the S 2-property of 8f1 , x and z have disjoint open neighbourhoods

Vi and Vi and, with the exception of a finite number of k, xnk c V1 , and, discarding the closures of the remaining xk from V2, we obtain a neighbourhood of z not intersecting F. In view of F n Gm+iCY) = 0 of course y nf(F) = 0, Y - f(F) again contains unk fork large enough, althoughf(xnk) Eunk; a contradiction. As there is a development for W2 , by (8.4.5) (i) we have to observethat 8!2 is fully normal. However this follows frorn (8.3.16) and (8.3.15)by (8.3.52). If W1 is rnetrizable, and thus a Ti-topology, then by (7.4.14) $ 2 has the sarne property and y = y, rnar f- 1(.Y) = rnar f-1(y). I 8.4.g. Embedding into product spaces. In Chapter 7, a series of theorems was proved showing that certain types of spaces can be topologically embedded into products of suitably chosen spaces. A theorem of this character will be proved here for rnetrizable spaces. Let us first introduce the spaces needed as factors of the products in question:

(8.4.13) Let I be a given index set and denote by A 1 the set consisting ofan element w and of all pairs of the form (s, i) where O < s ~ l and i EI. With the notation (0, i) = w for i EI, define

.

.

p((s,1), (t,J)) =

{Iss +- tt I

if if

i = j, i # j.

Then p is a metric on A 1 and induces a topology W1 • Proof. The definition of p is unambiguous even if triangle inequality needs to be proved. Now p((s, i), (u, k))

~

+

p ((s, i), (t,j))

s =

0 or t

=

0. Only the

p((t,j), (u, k))

means either

Is

- u

I ~ Is

- t

I + It

- u

I,

or

Is

- u

I ~ s + 21 +

s

+

u

s

+

u ~ s

u,

or ~

Is - t I

+t+

u,

or

+ t + It -

u

I,

or

s

+

u

~

s

+ 2t +

u,

according to whether i = j = k, or i = k and i # j, or i = j # k, or i # j == k, or i # j, j #, k, i # k, and all these inequalities are clearly true. I The rnetrizable space [AI> $ 1 ] is called asteroid with index set I. Now we can prove the following embedding theorem: (8.4.14) If [E, W] is a metrizable topological space, then there is an asteroid [A1, 8!1 ] such that [E, W] is homeomorphic with a subspace of the product space

(8.4.14)

8.4.g.

EMBEDDING INTO PRODUCT SPACES

381

[A}", 8ffl More precisely, if ~ = {Bk: k EK} is a base for gr, then a subset I c c K x K may be chosen as the index set of the asteroid. Proof. According to (8.4.5), there is a er-discrete base Q:: = {C/ j E J} for gr_

Let us consider those pairs (k, k') EK x K for which there is a j E J such that c Cj c Bk., and denote by j(k, k') one of these j. Then the sets Dkk' = Cj(k,k') constitute a base for gr_ In fact, if x EE and Vis a neighbourhood of x, then there are k' EK with x E Bk, c V, j E J with x E Cj c Bk, and finally k EK with X E Bk C Cj. Hence X E cj(k,k') C V. Thus there is a base ® for gr which is er-discrete (being selected from Q::) and whose elements can be equipped with indices taken from a subset I of K x K.

Bk

OCJ

Assume ® = {S;: i EI}, I=

LJ

In, and let ®n = {S;: i E In} be discrete.

1

For m EN, n EN, i E In, let U; be the union of those elements Sj E ®m (j E Im) for which Sj c S;. By (8.3.3) and (8.3.7), we have U; c S;, hence there is, by (2.5.40) and (4.2.2), a gr_continuous function J;: E -+ R such that O ~ f;(x) ~ 1 for x E l,f;(x) = 1 for x EU; andf;(x) = 0 for x EE - S;. Let us define a map .Qmn: E -+ A1 satisfying gmn(x) = (f;(x), i)

if XE S;, i E In,

gmn(x) =

if XE E -

W

LJ S;. iEln

The definition is unambiguous since S; n Sj = 0 if i, j E Im i =fa j. Furthermore gmn is (gr, gr1)-continuous. In fact, gmn I S; is clearly (gr I S;, gr1)-continuous for i E In, and gmn I Tn is (gr I Tm 8l1)-continuous for iEln

since x ES; - S; implies /;(x) = 0, gmn(x) = (0, i) = w so that gmn I Tn is constant. The system of the sets S; (i E /n) and Tn is a closed locally finite cover of Eby (8.3.2), (8.3.5), and (8.3.6), thus g;;,!(F) is closed by (8.3.7) for each closed subset F c A 1 . Now if x EE and V is a 8f-neighbourhood of x, then x E S; c V for some i E In, n EN, and with gr regular, also XE sj C S; for some j E Im, m EN. Hence x E U;,f;(x) = 1 and, with the notation y = gmix) = (I, i), clearly X

Eg;;,!(W) CS; CV

for the ball W with centre y and radius

~ . By (7.1.55), [E, 8f] is homeomorphic with

a suitable subspace of [A}" xN, 3JflxN]. Finally the latter space is homeomorphic with [Af, gr~] by (7.1.40). I Let us note that, conversely, each subspace of a product space [A}", 8f}"] is metrizable by (7.3.33).

8.4.h. Exercises. 1. Give in the space [R, $] (a) a regular base;

382

8.4.

METRIZATION THEOREMS

(b) a point-regular but not regular base;

(c) a non-point-regular base. [The system of intervals ( : , m : 2 ) (m = 0, intervals and the sets

± I, ± 2, ... ,

n EN}; the same

(o, : ) u (n, + CtJ) (n EN); the intervals with rational end-

points.] 2. Show that there is no point-regular base for i. 3. Show that if 58 is a (point-) regular base in the topological space [E, gr] and 0 =I- Ac E, then 58 (n) {A} is a base with the same property for gr I A. 4. Show that $+ is perfectly normal. [If G is $+-open, then, in view of the Lindelof property of $+ I G, G = (c): {x: f(x) = O} = ~ x: I f(x) I (e): The closedness of mar (B - A) implies A = A. (e) => (f): Obvious. (f) => (g): A = mar G = mar (B - G). (g) => (a): If Fis closed then mar Fis also closed, and rare too because x Eint mar F = int mar F = int(F - int F) c int F = int F

would contradict the equality mar F = F - int F.

I

(9.1.6.) Let [B, Sf] be a topological space, 0 =f. B0 c B, AC B 0 • If A is Sf i B 0 rare, then it is Sf-rare as well. Conversely if A is Sf-rare and B1 C B0 C £ 1 where B1 is Sf-open, then A is Sf I B 0 -rare. Proof. Suppose that A is Sf I B0-rare and int A- = G =f. 0. Then G n B0 =f. 0 as G c AC £ 0 and G n B0 CA n B0 ; accordingly the Sf I B0-closure of A contains a non-empty Sf I B0-open set although A is Sf I B0-rare. Hence int A = 0. Let now A be Sf-rare and B0 Sf-open. If there were a Sf-open G such that 0 =f. =f. G n B0 C A n B0 , then G n B0 being Sf-open, int A =f. 0 would hold; therefore the Sf I B0-interior of the Sf I B0-closure of A would be empty, i.e. A would be Sf I B0-rare. Finally if B1 c B0 c £ 1 and B1 is Sf-open, then A c (An B 1) u (B0 - B 1). Here A n B1 is Sf I Bi-rare according to the preceding result and then it is Sf I B0rare by the first part of the statement. On the other hand, B0 - B1 is the difference of the Sf IB0-closure of B1 and Bi, and hence identical with the Sf IB0-boundary of the Sf I B0-open set B1 and so it is Sf I B0-rare by (9.1.5). The same can be said about A by (9.1.1) and (9.1.3). J (9.1.7) If every point of A has a neighbourhood V such that V n A is rare, then A is rare as well. Proof. Otherwise let x E int A, and let V be an arbitrary neighbourhood of x. With the notation G = int V n int A, G is open, x E G, and G c A implies G c A n G c A n V, thus x E int A n V, and A n V is not rare. I 9.1.b. Meagre sets. The union of countably many rare sets is not always rare; e.g. in the space [R, $] the set Q, non-rare because Q = R, is the union of count• ably many sets consisting of a single point and being therefore evidently rare. Applying this reasoning to the space [Q, $ I Q], we can see that even the whole space can be the union of countably many rare sets. 25 Akos Csaszar

386

9.1.

(9.1.8)

RARE AND MEAGRE SETS

Let us introduce therefore the following terminology: a set is said to be meagre (or of the first category) in a topological space if it can be written as the union of countably many rare sets. (9.1.8) Every rare set is meagre.

I

(9.1.9) A subset of a meagre set as well as the union of countably many meagre sets is meagre. I It is important but much more difficult to prove the theorem corresponding to (9.1.7): (9.1.10) In a topological space [E, 8!], a set Ac Eis meagre if! every point x E A has a neighbourhood V such that V n A is meagre. Proof. The necessity of the condition is evident by choosing V = E. Suppose then that the condition is fulfilled. Consider all open sets G -=I= 0 for which G n A is meagre and the disjoint systems {G;: i EI} consisting of them (that is those for which i -=I= j implies G; n Gj = 0). Of course the collection of these systems is inductive so that, as a consequence of the Kuratowski-Zorn lemma (1.1.28), there is a maximal one among them. Let {G;: i EI} be such a one and put G = U G;. iE/

Then Ac G, for if x EA - G then by (9.1.8) x would have an open neighbourhood V such that V n A would be meagre and V n G = 0; but this contradicts the maximality of the system {G;: i EI}. Hence A - G c G - G = mar G, thus A - G is rare in view of (9.1.5) and (9.1.1). 00

As A n G; is meagre for every i, say A n G; = U A;m where A;n is rare, therefore n=l

with the notation An = U A;m Ann G; = A;n is rare for every i; thus by (9.1.7) iEl

An is rare as well, since every point of An lies in one G; and then G; is a neighbourhood of this point such that it intersects An in a rare set. Now An G = U (An G;) = 00

=

U U A;n iEJ n=l

iE/

00

=

U A~ shows

that An G is meagre; but we have seen that A- G is

n=l

rare and thus A is meagre as well.

II

As a consequence of (9.1.6) we can still say: (9.1.11) Let [E, 8!] be a topological space, 0 -=I= E 0 CE, AC E 0 • If A is 8T I E 0 meagre, then it is 8!-meagre as well. Conversely if A is 8!-meagre and E 1 c E 0 C .E1 where E 1 is 8!-open, then A is 8T I E 0 -meagre. II 9.1.c. Exercises. 1. Show that, in the space [R, $], A is rare iff it is bounded below. 2. Give an example of a setE, of topologies § 1 and § 2 onE, and further of subsets A, Be E such that 8J1 < 8!2 , A is Wi-rare without being 8J2-rare, Bis 8J2-rare but not Wi-rare. [E= R, 8J1 = ~ 8J2 =$,A= (0, 1), B= {-n:nEN}.]

(9.2.1)

9.1.c.

EXERCISES

387

3. Show that the sets rare with respect to SFE are identical with the finite sets. 4. Consider the topologies@ and@+ on Rand show that a set Ac R is @+-rare iff it is @-rare. [The @+-closure of A is a subset of its @-closure and the difference of these two closures is countable; the @+-interior of a set is empty iff its @-interior is empty; if Bis @+-closed, Mis countable, a < b, and (a, b) c Bu M, then (a, b) c B.] 5. Show that, in the space [R, @], every countable set is meagre and give an uncountable rare set. [The Cantor set.] 6. Let f be a function defined on R2 which is partially continuous (i.e. @x ®and ® x@-continuous where ® denotes the discrete topology of R). Show that (a) If f is not @2-continuous at the point z0 = (x0 ,y0 ), then there are rational numbers q, ,r. and an n EN such that I x - x 0 I < _!_ n implies q < f(x, y 0) < r, but in every @2-neighbourhood of z 0 the function f takes on values either smaller than q or greater than r; (b) The set Aq,r,n of the points z 0 having the above property is @2-rare; (c) The points at which f is not @2-continuous constitute an @2-meagre set. 7. Let [X, 8fi] and (Y, 8l2 ] be topological spaces, fa 8l1 x®y- and ®x x 8f2continuous function. Show that the points at which f is not 8f1 x 8f2-continuous constitute a 8T1 x 8f2-meagre set if 8T1 is either metrizable or an M 2-topology.

9.2. BAIRE SPACES 9.2.a. Baire spaces. (9.1.9) shows that the meagre sets can be considered "small" in some sense although - as we have seen - it can occur that the whole space is meagre. The meagre sets have the very character of "small" sets in those spaces in which the whole space or a (non-empty) open set cannot be meagre. We call a Baire space a topological space in which only the empty set is a meagre open set; the topology of a space of this kin,d is called a Baire topology. Any discrete space is an example of a Baire space (for only the empty set is meagre) as well as the space [E,SFE] if Eis an uncountable set (for the rare sets are finite and thus meagre sets are countable, while the non-empty open sets are uncountable). More interesting examples will be shown later.

(9.2.1) In any topological space [E, 8T] the following statements are equivalent: (a) 8T is a Baire topology; (b) If A is meagre then int A = 0; 00 (c) If Fn is closed and int Fn = 0, then int ( U Fn) = 0; 00

(d) If Gn is open and dense, then

·1

nGn is dense as well. 1

Proof. (a)=> (b): If A is meagre then by (9.1.9) int A is meagre and open, thus empty. 25*

9.2.

388

(9.2.2)

BAIRE SPACES 00

(b) => ( c): If Fn is closed and int Fn = 0, then Fn is rare, thus

UFn

is meagre.

1

(c) => (d): If Gn is open and dense, then E - Gn is closed and int (E - Gn) = 00

00

00

1

1

nGn) = 0 and n Gn is dense.

= 0, thus int ( LJ (E - Gn)) = int (E 1 00

(d) => (a): If G is open, G = LJAm and An is rare, then Gn = E - An is open 1

00

and dense, thus

n(E 1

00

00

dense, int ( UAn) = 0; accordingly

1

0.1

G C LJAn implies G =

00

UAn is

An) = E -

1

1

If [E, Sf] is a Baire space, 0 =I= E 0 CE, and E0 is either Sf-open or Sf-dense and Eo = nGn where Gn is Sf-open, then gr I Eo is a Baire topology. 1 Proof. Consider a sequence of SJ I E 0 -dense, SJ I E 0-open sets Hk (k EN). Then (9.2.2)

Cl)

Hk = Hi n E 0 where Hi is SJ-open. We can suppose that E -

E0 c Hi since

Cl)

Then clearly Hi is SJ-dense for k EN. Hence 00

co

1

1

nHi is SJ-dense and if E 1

0

is SJ-open 00

nHk = nHin E is clearly Sf I E -dense. If E is SJ-dense, E = nGm and Gn is Sf-open, then nHk = nHin nGn is Sf-dense and a fortiori Sf I E -dense.

then

0

0

00

00

00

1

1

1

0

0

1

0

By (9.2.1) SJ I E0 is a Baire topology in both cases.

I

(9.2.3) If [E, 8!] is a topological space, E0 c E is Sf-dense and Sf I E0 is a Baire topology, then Sf has the same property. Proof. If Hk is SJ-dense and SJ-open for k E N, then Hk n E 0 is clearly SJ I E 0oo

dense and gr I Eo-open. Hence is a fortiori SJ-dense. I

Cl)

nHk n Eo is gr I Eo-dense, thus SJ-dense, and nHk 1

1

9.2.b. Base-compact and hypocompact spaces. We shall consider a series of classes of spaces which will turn out to be in a close relation with Baire spaces. A topological space (or a topology) is said to be base-compact if there is a base 58 such that, for any centred system c c 58, the elements of c have a common limit point. The space (or the topology) is said to be hypocompact if the same is true for filter bases instead of centred systems, in other words if there is a base 58 such that every filter base t c 58 has a cluster point. Countably base-compact and countably hypocompact spaces are defined in a similar manner, with the restriction to countable centred systems and countable filter bases, respectively. Finally a space is said to be weakly base-compact or weakly hypocompact if there is a sequence (58n) of bases such that Bk E 58nk' 00

n1 < n2 < ... implies respectively.

n Bk =I= 0 for a centred system or a filter base {Bk: k EN} 1

(9.2.7)

9.2.b.

389

BASE-COMPACT AND HYPOCOMPACT SPACES

(9.2.4) The following implications are valid;

= hypocompact

Base-compact

~

~

countably base-compact ~

= countably hypocompact ~

weakly base-compact

= weakly hypocompact. I

The connection of the classes considered here with Baire spaces is given by the following theorem: (9.2.5) A weakly hypocompact regular space is a Baire space. Proof. Let(){\) be a sequence of bases as in the definition and (Gn) a sequence of dense open sets. If G #- 0 is open, then take x 1 E G n G1 and select B1 E Q3 1 such that

Then let x 2 E B1 n G2 and B2 E Q3 2 be such that

Similarly there exist

Xn

and Bn E )Sn with

Clearly {Bn: n EN} is a filter base so that 00

0 #-

nBn c 1

00

Gn

nGn 1

00

and

nGn is dense. Hence (9.2.1) furnishes the statement. I 1

(9.2.6) Every locally compact S 2 -space is base-compact and hence a Baire space. Proof. Let Q3 be the system of all open sets having compact closures in the space [E, 81]. By (5.3.34) and (5.3.4) Q3 is a base. If cc Q3 is a centred system and B0 E c, then

{B n B0 : BE c} is a centred system in the compact space [B0 , 81 I B0 ], hence

n {B: B E c}

=

n {B n Bo:

B E c} #- 0 .

By (5.3.54) 81 is regular and (9.2.5) reveals that it is a Baire topology.

I

(9.2.7) The topology of a complete pseudo-metric space is weakly hypocompact.

9,2.

390

(9.2.8)

BAIRE SPACES

Proof. Let )Sn denote the system of all open sets having a diameter less than

_!__ n

Clearly )Sn is a base. If r = {Bk: k EN} is a filter base and Bk E )Sn"' n1
••• , 0 =/a Bk E ~nk' n1 < n 2 < ... imply '

n1 Bk =fa 0.

The space is countably hypocompact if! the same condition holds for ~n = ~. Proof. Since a decreasing sequence of non-empty sets is a countable filter base,

the necessity is obvious. Suppose that the above condition is satisfied for a sequence C~n) of bases and let t = {Bk: k EN} be a filter base such that Bk E ~nk• n1 < n2 < . . .. Let m1 = 1 and, if mk is defined, let qk be an index such that mk

n Bi is not contained in Bqk and select mk+

1

satisfying

i=l

mk Bmk+l c Bqk n

nBi.

i=l

The construction stops if, for a k,

mk

nBi =fa 0 is contained in every element of t

i=l

in which case t clearly has a cluster point. If the construction never stops then

392

9.2.

(9.2.11)

BAIRE SPACES

clearly m1 < m2 < ... and Bm, => Bm, => ... so that Bmk E IBnmk' nm, < nm, < ... , and 00 00 0 =I nBmk c n.B; 1

1

since Bmk+ 1 c B; for i ;£ mk. Hence the space is weakly hypocompact and if, moreover, IBn = IB for n E N, then it is countably hypocompact. I In connection with heredity properties we can say the following: (9 .2.11) An open subspace of a ( countably or weakly) base-compact or hypocompact regular space has the same property. Proof. Let IB be a base (or (IBn) a sequence of bases) in the space[E, 8J] satisfying the respective condition. If 0 =I E 0 c Eis 8J-open, let IB' (or IB~) consist of those sets BE IB (or BE IBn) for which B c E0 • Then IB' (or IB~) is a base for 8J I E0 and IB' (or the sequence (IB~)) assures the required property of [E0 , 8J I E 0 J. I

A theorem analogous to the second part of (9.2.2) can be stated for weakly base-compact spaces: (9.2.12) If Gn is open and dense in the weakly base-compact regular space [E, 8!], 00

then

8)

I

nGn is weakly base-compact as well. 1

Proof. By (9.1.3) we can assume that G1 => G2 => .... Let (IBn) be a sequence of bases for 8J satisfying the condition in the definition of weak base-compactness, 00

and let IB~ consist, with the notation E0 = n Gm of those sets B n E0 for which 1

BE IBm BC Gn- Then IB~ is a base for 8J I E 0 • Now if

and the system 00

is centred, then {Bk: k EN} is a fortiori centred, hence there is an x En Bk and 1 clearly 00

XE nGllk = 1

00

nGil = Eo. 1

On the other hand E 0 is 8f-dense by (9.2.4) and (9.2.5), hence Bk n E 0 => Bk so that the 8J I E 0-closure of Bk n E 0 , i.e. Bk n E 0 n E 0 , contains x for k E N. I 9.2.c. Product theorems. The product of two Baire spaces need not be a Baire space. However, under some restrictions, it is possible to establish positive results. Let us notice first:

(9.2.13) Let [X, 8fi] and [Y, 8!2 ] be two topological spaces and let the latter be an M 2-space. If Ac Xx Y is 8J1 x 8!2-rare, then there is a 8!1-meagre set SC X such that Ax = {y: (x, y) EA} is 8!2 -rare for x EX - S.

9.2.c.

(9.2.16)

393

PRODUCT THEOREMS

Proof. Let {Bn: n EN} (Bn =I 0) be a base for g[2 and Cn the set of those points x EX for which Bn c Ax. Then Cn is gT1-rare, for if G is gT1-open and G c En then by (7.1.27) G xBn c En xBn = en xBn and as (x, y) E en xBn implies (x, y) E {x} xAx C {x} xAx CA, i.e. Cn xBn CA, therefore G xBn CA which 00

is possible only if G x Bn == 0, i.e. if G = 0. For the gr1-meagre set S = .

U Cn 1

it

now holds that if x EX - S then Ax is gT2-rare since otherwise Bn c Ax would hold for some n. I (9.2.14) Let [X, g[1 ] and [Y, g[2 ] be topological spaces and the latter be an M 2 space. If Ac Xx Y is gr1 x gr2-meagre, then there is a grrmeagre set Sc X such that Ax = {y: (x, y) EA} is gr2-meagre for x EX - S. 00

Proof. Let A= LJAm where An is gr1 xgr2-rare; referring to (9.2.8) we see 1

that Sn is a grrmeagre set such thatAnx = {y: (x, y) E An}is gr2-rareforx EX - Snoo

Then S =

USn is grrmeagre and if x EX 1

00

S then Ax =

UAnx is gr2-meagre. I

n=l

(9.2.15) If [X, gril and [Y, gr2 ] are Baire spaces and the latter is an M 2 -space, then gr1 x gr2 is a Baire topology. Proof. If G =I 0 is gr1 x g[2-open, then for a suitable grropen set G1 =I 0 and a gr2-open set G2 =I 0 we have that G1 x G2 c G and together with G also G1 x G2 would be gr1 x gT2-meagre. As x E G1 implies {y: (x, y) E G1 x G2 } = G2, therefore if G2 were not §'2-meagre, then G1 would have to be grrmeagre in view of (9.2.9).

From this it would follow that G2 = 0 or G1 = 0 which is a contradiction.

I

(9.2.16) An arbitrary product of (countably or weakly) base-compact or hypocompact spaces has the same property. Proof. Let 5Si be a base ( or (5S;n) for n E N a sequence of bases) in the space [E;, gT;] satisfying the condition of the respective definition. Let 5B' (or 5S~) consist of all Cartesian products X A; where A; E 5B; (or A; E 5S;n) for a finite number of iEI

indices i E J and A; = E; for the remaining indices. Then 5B' (or 5S~) is a base for gr = X gri· If c is a centred system or a filter base taken from 5B' (or c = {Ck: iE/

k EN}, Ck E 5S~k' n1 < n2 < ...), then the ith factors of the elements of c constitute, for each i EI, a centred system or a filter base respectively, the elements of which either coincide with Ei or belong to 5B; (or to 5S;nk). Hence these ith

factors have a common limit point x; EE; and then x = (x;) is a common gr-limit point of all sets taken from c. I

In particular, the product of weakly hypocompact regular spaces is a Baire space by (9.2.5); the same conclusion holds by (9.2.4), (9.2.6), and (9.2.7) if each factor is either locally compact and (SJ or complete pseudo-metric. 9.2.d. Applications. By using Baire's theorem the following theorem can be proved, furnishing a useful counter-example for various purposes:

9.2.

394

(9.2.17)

BAIRE SPACES

(9.2.17) The space [R2, (~+)2] is not normal. Proof. Consider the line with equation x + y = 0 and let A and B be respectively the sets of the points with.rational and irrational coordinates on it. Evidently both A and B (and even any subset of the line mentioned) are (~+)2-closed. If 00

G and Hare (~+)2-open sets containing A and B respectively, then A =;=

1

00

B

LJ Am

= LJ Bm where An and Bn denote the sets of those points (x, y) EA and (x,)) EB

r

1 respectively for which Qn(x, y) = _x,x

+

n)1· x ry,y + nlJ , or Qn(x, y) CH. CG

If now Cn and Dn denote the projections on the x-axis of An and Bn respectively and X denotes the closure with respect to ~ ( !) of the subset X of the x-axis, then 00

according to R =

00

LJ1 en u LJ1 15n and

(9.2.1), int

en =fa 0 or int Dn =fa 0 for at least .

one n (int is also understood with respect to the topology ~) since ~ is a Baire topology by (9.2.8). Suppose that a < b and (a,b) c Choosing then an irrational number .a < x 0 < b, it holds that (x0 , -x0) E Bm for some m, thus Qm(x0 , -x0) c H. On the other hand numbers xE Cn can be found arbitrarily near to

en-

x 0 while Ix - x 0 I < min (

!,!)

implies Qn(x, -x) n Qm(x0 , -x0) =fa 0 so that

by QnCx, -x) c Git follows that G n H =fa 0.1 As by (2.5.41) ~+ is a T5-topology and a fortiori a T,.-topology, therefore (~+) 2 is a T,.-topology as well by (7.1.37). Hence the space [R2, (~+)2] is an example of a non-normal Tykhonov space. At the same time it shows that the product of normal (moreover of completely normal) spaces is not necessarily normal. From (2.4.21) it is also known that the space [R, ~+] is a Lindelof space, thus a paracompact T2 -space by (8.3.19) and (8.3.15); therefore the product of paracompact T2 -spaces need not be paracompact and is not even necessarily normal. Further important applications of Baire's theorem and Baire spaces are the proofs of some existence theorems. That is, if we want to show the existence of a point, set, function, map, etc. with a given property, it is sufficient to prove that the points, sets, functions, maps, etc. which do not have the given property constitute a meagre set with respect to a suitable Baire topology; it follows from this that the set of those with the given property cannot be empty. We show two notable examples of this method. (9 .2.18) Let Un) be a pointwise convergent sequence offunctions continuous in the topological space [E, 3f], f(x) = limf,,(x) (x EE). Then those points of Eat which n-oo

f is not continuous constitute a meagre set; if 3f is a Baire topology, then the set of those points at which f is continuous is dense. Proof. If f is not continuous at the point x EE, then, for a suitable e > 0, in every neighbourhood V of x there are y, z E V such that If(y) - f(z) I > e. It is sufficient to show that the set D. of these x is meagre for in this case the set 00

U D!, meagre as well, contains all those x at which/is not continuous. 1

n

(9.2;18)

9.2.d.

Let now An be the set of those ,x EE f~r which i, j

~ : . As IJi -

395

APPLICATIONS

~ n

implies Ifi(x) - Jj(x)

I~

Jj I is continuous, the set of these x is closed for given i, j and Am Cl()

UI An follows from

since the intersection of these sets is closed as well. As E = ,

oo

the

Cauchy condition, therefore D. = U (An n D.) and it is enough to show that I

An n D. is rare for every n EN and e > 0. Suppose that G is open and x E G C Ann D •. Then G this Ifi(y) - fn(y)

If(y)

- fn(y)

implies

I~ 4

I ~ : . If

Ifn(y)

lf(y) -

8

- fn(x)

fn(y)

if y E G and i

~

C

An =

n, and then by letting i

---+

An and fq,m oo we obtain

now V is an open neighbourhood of x such that y E V

I < : , then, for

y, z E V n G,

lf(y) - f(z) I ~ I + lfn(y)-fnCx) I + I fnCx)-f,,(z) I+ lfnCz)-f(z) I
0 and 0< h ~ u, wh'l 1e if J

~

h ~ b - x then it will be bounded in view of the boundedness off We shall

. a funct10n . f,, continuous . . [a, b] , 1or c now prove that there 1s m whi chf(x+h)-f(x) h is not bounded for any given x with a ~ x < b and O < h ~ b - x, moreover that these functions constitute the major part of the functions continuous in [a, b]. In order to formulate the latter statement exactly, denote by Ethe set of all functions continuous in [a, b] and introduce the metric p(f,g)

= sup {Jf(x) - g(x) J : a~ x

~ b}

for f, g EE. We know from 1.3.c that [E,p] is complete. Hence the topology 8TP is a Baire topology by (9.2.8) and we shall show that, with respect to it, the functions

396

9.2.

. f(x f EE, for which

one a

~

+ h) h

(9.2.19)

BAIRE SPACES

f(x) . 1s bounded whenever O < h

~

b - x for at least

x < b, constitute a meagre set:

(9 .2.19) Let [a, b] c R, E be the set of functions continuous in [a, b] with the metric p(f, g) = sup { [f(x) - g(x) I: a ~ x ~ b} and A c E the set of those f for which at least one a ~ x < b can be found such that f(x + h) - f(x) . . h zs bounded for O < h ~ b - x. Then A zs §>meagre and as §"P is a Baire topology, E - A is §"P-dense. Proof. If/EA, then there are an a~ x< band an n EN such that a 1 - - and n

I f(x + hh -

f(x)

I~

n

(0 < h

~

~

x

~

b-

b - x).

Let An c E consist of those f which fulfil these two conditions for at least one x. 00

It is enough to see that An is rare with respect to the topology §"P as A = U Ant

First An is 8JP-closed. Indeed if fk E An (k EN) and p(fk,f) --+ 0, then, for a suitable xk, a

~

xk

~

b - _!__ and O < h n

~

I fixk + ~ -

b - xk implies fk(xk)

I~

n.

According to the Bolzano-Weierstrass theorem a suitable subsequence (xk) converges to an x which also fulfils a if already xk; < x [f(x

+ h,

~x~

b - _!__ and, whenever O< h n

~b

- x,

then

+ h) - f(x) [ ~ [f(x + h) - fk/x + h) I + + I fk/x + h) - fk/xk;) I + I fk;(xk) - f(xk;) I + + I f(xk) - f(x) I ~ p(f,fk) + n(x + h - xk) + + p(fk;,f) + If(xk) - f(x) I --+ nh,

as p(f,fk) --+ 0, x + h - xk;--+ h and by the continuity of f,f(xk) - f(x) --+ 0. Accordingly f EAnHence it will be proved that the set An is rare if we show that int An = 0, i.e. E - An is dense. For this purpose, let f EE, e > 0 be given. Using the uniform continuity off, consider a subdivision a = x 0 < x 1 < ... < xk = b of

(9.2.20)

9.2.e.

[a, b] such that xi-l

397

CECH-COMPLETE SPACES

~ x < y ~ x 1 implies lf(y) -f(x) I< : , and let p be the

function for which p(x1) = f(x1) (i = 0, ... , k) and p is linear on the segments [x1_ 1, xi]. Of course if x 1_ 1 ~ x ~ x 1, then If(x) - p(x) I ~ If(x) - f(x,) I + + If(x1) - p(x1) I + I p(x1) - p(x) I
f(x, 0) for n--> oo; (f) The sets A and Bin (d) cannot be separated with a Si-continuous function; (g) Si is not normal. 17. Let [E, Si] be the space defined in the preceding exercise. Show that (a) Si is base-compact; (b) Si is Cech-complete; (c) Si is neither locally compact nor metrizable. [The Si-closure of an element of Q3 coincides with its ~?-closure. The Si-closure of K(x, e) is not Si-compact. If Q\ is the system of all elements of Q3 with radius less than

2-, and a centred system c in E contains, for each n EN, a set Cn c n

Gn E @m

then the sets of c have a common @2-limit point p in E, and p is clearly a common Si-limit point if p = (x, y), y > O; if p = (x, 0) then the sets Gn are necessarily of the form K(p, en) and pis a common Si-limit point again.] 18. Show that the condition of regularity can be replaced in (9.2.5) by the following condition: if G :f. 0 is open then there is an open set H ":f 0 such that

iicG. 19. (a) (b) (c) (d)

Let [E, Si] be the non-regular T2-space defined on p. 97. Show that Si is base-compact; [E, Si] is a Cech space; Si satisfies the condition of the preceding exercise; [E, Si] is a Baire space.

[If A={! :nEN} then(a,b) = [a,b] and (a,b) -A= [a,b] for the Si-

406

9.2.

BAIRE SPACES

(10.1.1)

closures. Let @n consist of the intervals (a, b) such that O~ [a, b ], b - a< _!__ and n of (-

!,!)-

A; if c is a centred system such that th~re is, for each n EN, en E c

with enc Gn E@m then the elements of c have a unique common ~-limit point;

= (- _!_, __!_J- A for each n and O is, a common 8i-limit point.] n n (R x(R - {O})) u (Q x {O}), 8i = @2 IE. Show that

if this is Othen Gn

20. Let E = (a). [E, 8i] is a Baire space; (b) E is not the intersection of countably many @2-open sets; (c) 8i is not weakly hypocompact; (d) A = Q x {O} is 8i-closed and 8i I A is not a Baire topology. [(9.2.8), (9.2.2), (9.2.26), (9.2.25).] 21. Let [E, p] be a complete pseudo-metric space. Construct a sequence D0 c D 1 c D2 c ... of sets such that Dn is maximal with respect to the property that x, y EDm x #= y implies p(x, y) ~ 2-n. Let ~ be the system of all balls S(x, 2-n) (x E Dm n = 0, 1, 2, ... ). Show that if tis a filter base taken from ~ then either t is Cauchy or there is in t a ball contained in any other member of r; hence we obtain a simple proof of the fact that 8TP is hypocompact. 22. Let E = R2, 8i = ~ x ®it, and D be the diagonal of the set R2 • Show that (a) D is 8i-rare; (b) {y: (x, y) ED} is not @R-meagre for x ER. 23. Let [X, 8ii] and [Y, 8i2 ] be Baire spaces, f a (8i1 x®y)- and (®x x 8i2)continuous function. Show that those points z E Xx Y at which f is (8i1 x 8iJcontinuous form a (8i1 x 8iJ-dense set if either 8i1 is anM2-topology or 8i1 is completely metrizable and 8i2 is weakly hypocompact and regular. [9.1.c.7, (9.2.15), (9.2.16), (9.2.7), (9.2.5).] 24. Show that there is an index set I such that @1 is not Cech-complete. [By (6.4.36) and (7.1.54) [Q, ~ I Q] is homeomorphic with a closed subspace of [RI, @1 ] fcir some I.] ' 25. Show that every almost compact space is base-compact. 26. Show that every countably compact space is countably base-compact. 27. Show that every completely regular pseudo-compact space is countably base-compact. [8.3.i.27,]

10. CONNECTED SPACES

10.1. CONNECTED SETS 10.1.a. Separated partitions. In the following we want to define exactly the intuitive statement that a topological space consists of "a single piece". Evidently we have to understand by this that the space cannot be decomposed into two "separated" subspaces; however as decomposing the underlying set of the space (consisting of at least two points) into two disjoint subsets we decompose the space as well into two subspaces, we require a suitable restriction for the subspaces in question. Recalling the fact that some properties of a topological space are "hereditary" for all subspaces, others only for closed or open subspaces, we shall consider as a "proper" decomposition only decompositions into disjoint and non-empty open or closed subspaces. In connection with this we note:

(10.1.1) Let [E, 8!] be a topological space, E = Au B. The following statements are equivalent: (a) An B = 0 and A and Bare closed,· (b) A and B are strongly separated; (c) A and Bare separated; (d) An B = 0 and A and B are open,· (e) A and B are disconnected. Proof. (a)=> (b) => (c): Obvious. (c) => (d): An jj =An B = 0 implies A = A, B = B, thus A = E - Band B = E - A are open. (d) => (e): Obvious. (e) => (a): If A c A1, B c Bi, A1 n B1 = 0, A1 and B1 are open, then A = A1, B = B1, thus A = E - B1 and B = E - A 1 are closed and An B = 0.1 A partition {A, B} satisfying any one of the preceding conditions is said to be a 8!-separated partition of E, namely proper if A # 0 #- B, or else trivial. We note further: (10.1.2) Let [E, 8!] be a topological space, Au B c E 0 c E, E 0 _# 0. A and B are 8!-separated iff they are (8T I E0 )-separated. Proof. An jj = An B = 0 and A n,B n E0 = An E 0 n B = 0 hold at the same time. I ,

be

Accordingly the partition C =Au B of the set Cc Ewill called &i"-separated if A and Bare 8!-separated or if A and' Bare 8T j, C-separated which is the· same by (10.1.2). By (10.1.1) and (10.1.2) it can be asserted:

408

10.1.

CONNECTED SETS

(10.1.3)

(10.1.3) Let [E, 8T] be a topological space, C =Au B c E, C 8T-closed (8T-open). A and Bare 8T-separated if! An B = 0 and both A and Bare 8T-closed (8T-open). J A separated partition of the set C is said to be proper or trivial according to whether its members are non-empty or at least one of them is empty. (10.1.4) If in the topological space [E, 8T] Cc Au B c E and A and Bare 8Tseparated sets then C = (A n C) u (B n C) is a 8T-separated partition of C. I (10.1.5) If in the topological space [E, 8T] both A 1 and B, and A 2 and B, are separated then A 1 u A 2 and Bare separated as well. Proof. (A 1 u A2) n Ji = (A 1 n B) u (A 2 n B) == 0, further A1 uA 2 n B = (A1 n B) u u (A 2 n B) = 0.1 10.1.b. Connected sets. Let us agree to call the topological space [E, 8T] (and the topology 8T) connected if E has no proper 8T-separated partition; similarly the set Cc E is said to be 8T-connected if it has no proper 8T-separated partition. It follows from (10.1.2):

(10.1.6) Let [E, 8T] be a topological space, C c E0 c E, E0 # 0. C is 8T-connected is 8T / E 0-connected, i.e., if C # 0, if! the subspace [C, 8T / C] is connected. I

if! it

For example, in any topological space the empty set and every set of one element is connected. A further important example for a connected space is the following: (10.1.7) The space [R, ~] is connected. Proof. Let R = A u B be a separated partition, A # 0 #: B. If a E A, b E B and for example a < b, then let c = sup(A n [a, b]).

Since by (10.1.1) A i(closed, therefore c EA. Hence c 0, [c, c + e] c A, c + e < b. There would follow c + e EA in contradiction to the definition of c. I (10.1.8) Let [E, 8T] be a topological space. The following statements are equivalent: (a) 8T is connected; (b) There is no open-closed set in [E, 8T] except 0 and E; (c) 0 #: C # /E, Cc E imply mar C # 0. Proof. (a) => (b): If Cc Eis open-closed, then E = Cu (E - C) is a separated partition. (b) => ( c): If mar C = 0, then C = int C = C, thus C is open-closed. (c) => (a): IfE = Au Bis a separated partition then A is open-closed by (10.1.1), thus A = A = int A and mar A = 0. I It is worth noting as a generalization of property (c) that:

(10.1.9) If in the topological sprice [E, 8T] C is connected and C n A # 0 # C - A then C n mar A # 0.

(10.1.14)

10.1.b.

CONNECTED SETS

409

Proof. Otherwise C c int A u (E - A) would hold and as the latter two sets are evidently separated and both of them intersect C as supposed, their intersection with C would give a separated partition of C by (10.1.4). I 10.1.c. Operations. Many operations can be given leading from connected sets to connected sets. (10.1.10) lf in the topological space [E, ST] C is connected and Cc C1 C C, then C1 is connected as well. Proof. If C1 = A u B is a proper separated partition, then C n A = 0 would imply A = 0 by C c B, C c B, A n jj = 0, so that C n A -:/= 0. In the same way C n B -:/= 0. Hence on the other hand by (10.1.4) C = (C n A) u (C n B) is a proper separated partition of C which is impossible. I

lf in the topological space [E, ST] C; is connected (i E/) and further 0, then C = LJ C; is again connected. iEl iEl Proof. If C = A u B is a proper separated partition, x En C;, and x EA, iEl then y EB implies y E C; for some i, further x E C; and hence C; = (A n C;) u (10.1.11)

n C; #

u (B n C;) is a proper, by (10.1.4), separated partition of C;. This is impossible.

I

(10.1.12) lf in the topological space [E, ST] one can find for any two points x, y a connected set Cxy such that x, y E Cxy, then the space is connected. Proof. Fix a point x EE, x En CX.J' E = LJ Cxy• and apply (10.1.ll). I YEE

YEE

(10.1.13) If E = Au Bu C in the connected topological space [E, ST], A and B are separated, C is connected, A n C # 0, then A u C is connected as well. Proof. Suppose that A u C = Mu N is a proper separated partition. If C n M # # 0 # C n N, then by (10.1.4) C would have a proper separated partition so that e.g. C n M = 0, Cc N. Then A :::> Mand as A and B are separated, Mand Bare necessarily separated as well. By (10.1.5) E = Mu (Nu B) would be a separated and, in view of M # 0, N # 0, a proper partition which is impossible. I (10.1.14) lf in the topological space [E, ST] A u Band A n Bare connected and both A and Bare closed (open), then A and B are connected. Proof. By (10.1.6) it may be supposed that Au B = E. Then E = (A - B) u u (B - A) u (A n B); here the members are disjoint, E and A n B are connected, A - B and B - A are separated since A - B n (B - A) c A n (B - A) = =An (B - A)= 0 and in the same way (A - B) n B - A = 0 (or if A and B are open then A - B n (B - A) c E - B n (B - A) = (E - B) n (B - A) = 0 and in the same way (A - B) n B - A = 0). Hence by {10.1.13) A = (A - B) u u (A n B) and B = (B - A) u (A n B) are connected as well. I

It follows at once from the definition that a space homeomorphic with a connected space is again connected. This can be essentially improved:

410

10.1.

(10.1.15)

CONNECTED SETS

(10.1.15) A continuous image of a connected topological space is again connected. Proof. Let [X, W1 ] be connected, f: X-+ Y (Wi, W2)-continuous, f(X) = Y. If Y =Au B is a proper separated partition, then X = f- 1(A) uf-1(B), f- 1(A) and f-1(B) are open and non-empty, and further

This is impossible.

I

With the help of these results we can obtain all connected sets in the space [R, &\]:

(10.1.16) In the topological space [R, $] a set is connected if{ it has the form (a,b), (a,b], [a,b), [a,b] (a~ b, a,bER), or (-CXJ, +CXJ), (-CXJ, a), (-CXJ,a], (a, + CXJ), [a, + CXJ) (a ER). Proof. It is known already from (10.1. 7) that ( - CXJ, + CXJ) is connected, and (a, a) = (a, a] = [a, a) = 0 and [a, a] = {a} are also connected. If a < b then the interval (a, b) is the image of R by the continuous map given by the fora+ b b-a x . . d h b ( mula y = - - -

2

+--- - - - 2

l+JxJ

hence 1t 1s connecte . T us y 10.1.10),

(a, b], [a, b) and [a, b] are connected. As the map y

1

= a - 1 + - maps (0, 1) X

onto (a, + CXJ ), therefore (a, + CXJ) and by (10.1.10) [a, + CXl) are connected as well. The :inap y = -x carries (-a, + CXl) and [-a, + CXl) over (- oo, a) and (- CXJ, a] respectively so that these are connected too. On the other hand if C c R does not belong to any of the types listed above, then we can find an a, b, c such that a< b < c, a, c E C, b ~ C. Now ( - oo, b) and (b, + oo) are separated and both intersect C so that by (10.1.4) C has a proper separated partition. I (10.1.17) If [E;, 8T;] is a connected topological space for i is again connected. Proof. Choose and fix a point X; EE; for every i E I and let x

EI, then 8T = X8ri iEI

=

(xi)

E X Ei = E. iEI

Denote by C the set of those points y = (Y;) EE only finitely many coordinates of which are different from the corresponding coordinates of x. Evidently C = E so that by (10.1.lO)it needs only be shown that C isW-connected. However this follows by (10.1.11) from the fact that, to every point y E C, we can find a connected set CY c C for which x, y E CY. In order to prove this, let y EC, say Yi = X; whenever i EI - {i1, ••• , in}. Then fork= 1, ... , n, let Ak be the set of those points z = (z;) for which z;1 = = xi; if j < k, zi1 = Yii if j > k, z;k EE;k is arbitrary, finally z; = X; if i EI - {ii, ... , in}. Sipce by (7.1.31) 8T I Ak is homeomorphic with W;k, then;!fore Ak is W-connected. Furthermore y E Ai, x E An and Ak n Ak+i =I 0 for every index k = 1, ... , n - 1 since, with the notation z; = xi if i EI - {ii, ... , in} or i = t, j ~ k and zi == Yi ifi= ij, j ~ k + 1, we have z = (zi) EAk n Ak+l' Therefore

(10.1.22)

10.1.d.

411

COMPONENTS

n

using repeatedly (10.1.11) the sets Ai, A1 u A2, n

CY= LJAk c C will do. 1

••• ,

UAk will be connected

and

1

I

10,1.d. Components. We understand by the components of the topological space [E, Sf] the maximal connected sets of the space, that is those sets C c E which are connected, and if C c C1 c E, C1 is connected, then C1 = C. It follows at once from (10.1.10) that:

(10.1.18) Every component of a topological space is closed.

I

(10.1.19) Two different components of a topological space are disjoint. Proof. If C1 and C2 are two components and C1 n C2 =/: 0, then by (10.1.11) C1 u C2 is connected as well; thus C1 = C1 u C2 = C2 . I (10.1.20) Any topological space is equal to the union of its components. Proof. In the space [E, Sf] the union of all connected sets containing x EE is connected by (10.1.11) and is evidently a maximal connected set" so that any point x E E is an element of some component of the space. I By (10.1.19) and (10.1.20) the components of every topological space give a partition into disjoint subsets of the space. We understand by the components of some subset 0 =/: A c E of a topological space [E, Sf] the components of the subspace [A, 3i" I A]. (10.1.21) If C is connected in the topological space [E, Sf], then C is a subset of some component of the space. Proof. If C1 is a component of the space and C n C1 =/: 0, then by (10.1.11) Cu C1 is connected as well; thus Cu C1 ~ Ci, C c C1 • I 10.1.e. Continua. A compact connected set (in a topological space) is said to be a continuum. The following remarkable theorem holds for continua:

(10.1.22) In an S 4-space the cluster points of a filter base consisting of continua constitute a (non-empty) continuum. Proof. Let [E, §J be an S4-space and f a filter base in E consisting of continua. By (5.3.9) the set C of the cluster points of f is non-empty, and by (5.2.24) C = n{K: KE f} is compact by (5.3.4) as a closed subspace of .K0 (where K0 E f is arbitrary) which is compact by (5.3.20). Let C=A u B be a proper separated partition. Then both A and Bare closed by (10.1.3), and thus there are open sets G and H such that A c G, B c H, G n H = = 0. If there were no set KE f such that Kc G u H, then the sets K - (Gu H) =/: =/: 0 would constitute a filter base consisting of compact sets which would have therefore a cluster point although {.K...,.. (Gu H): KE f} = {.K: KE f} - (Gu H) = 0. Hence Kc Gu Hfor some Kin f and as G and Hare separated, further 0 =/:Ac Kn G, 0 =/: B c Kn H, therefore K =(Kn G) u (.Kn H) is a proper separated partition by (10.1.4) although Kis connected in view of {10.1.10). This contradiction shows that C is connected. I

n

n

412

10.1.

CONNECTED SETS

(10.1.23)

From here we obtain by (5.2.24): (10.1.23) If in an S 4-space K 1

::)

K 2 ::) K 3

::) .••

is a sequence of non-empty

00

continua, then

nKn :f= 0 is a continuum as well. I 1

10.1.f. Exercises. 1. Show that (a) Every subspace of the space [R, $] is connected; (b) In any Ti-space consisting of more than one point there is a non-connected subspace. 2. Show that in the space [E, §:E] a set is connected iff it is either empty, or it has only one element, or it is infinite. 3. Show that in a zero-dimensional T0-space only 0 and the sets of one element are connected. 4. Let Ebe an ordered set, gr the order topology of E. Show that gr is connected iff the ordering of Eis complete and there is no jump in E (i.e. if a, b EE, a < b implies that there is a c EE such that a < c < b). [If a< band there is no element c with a< c < b then E =(+-,a] u [b, -+) is a proper separated partition; if Ac E, a EA, bis an upper bound of A but there is no smallest upper bound, then the upper bounds of A constitute an open-closed set containing b but not containing a. The connectedness of gr follows from the conditions in the same way as that of&.] 5. Lf't E = R x I, and, for (xi, Yi), (x2 , y 2) EE, let (xi, Yi) < (x2 , y 2) if x 1 < x 2 or x 1 = x 2 and Yi < y 2 • Show that the order topology of Eis connected. 6. Show that a T"-space which is connected and has at least two points cannot be countable. [There is a continuous function whose range is uncountable.] 7. Show that if [Y, gr] is a connected topological space and/: X-+ Y is a surjection then 1- 1(8!) is connected as well. 8. Show that the supremum of connected topologies need not be connected. [E = {O, l}, the grropen sets are 0,E, {O}, the 'lf2-open sets are 0,E, {1}.] 9. Show that the space of example 9.2.g.20 is (a) rim-compact; (b) not locally compact; (c) not zero-dimensional. 10. Let E c R2 be the set of those points (x, y) for which x is rational, 8f = = &2 I E. Show that (a) If z0 = (x0 , y 0 ) EE and Vis a gr_neighbourhood of z0 such that V c [x0 - 1, x 0 + 1] x [y0 - 1, Yo+ 1], then, for a suitable J > 0 and every rational x E [x0 - J, x 0 + J], there is a 8f-boundary point of Vwhose first coordinate is x; (b) The gr_boundary of V with the above property cannot be gr_compact; (c) gr is not rim-compact. 11. LetE c R.2 be the set of those points (x, y) for which either x or y is rational. Show that .$2 I E is connected.

(10.1.23)

10.1.f.

413

EXERCISES

12. If a, b E Rm, the set of the points (a1 + t(b1 - a1), . . . , am + t(bm - am)) is called the segment ab where a = (a1 , . . . , a~, b = (bi, ... , bm) and t E I. Show that (a) Any segment ab is connected in [Rm, li!,mJ; (b) If x ES(a, 8) then the segment ax is a subset of S(x, 8); (c) S(a, 8) is connected. 13. Call in Rm the union of the segments a0ai, a1a 2 , ••• , an_ 1an a polygon. Show· that (a) In [Rm, li!,mJ every polygon is connected; (b) If G c Rm is open and a EG then those points x EG for which one can find a polygon in G containing a and x constitute an @m I G-open-closed set; (c) The open set G c Rm is connected iff, for a, b E G, there is a polygon in G containing a and b. 14. Show that the components of the space [E, 8fJ in 10. are sets of the form {x}xR (xEQ). 15. Show that if, in the topological space [E, 8fJ, Cc Eis connected and openclosed, then it is a component of E. 16. Show that if, in the topological space [E, 8fJ, C c A c Band C is a component of B, then it is also a component of A.

l;, Let Q = [O,lJ x [O,lJ c R 2 • Remove from Qthe open square (

I

!,:) !,:), x(

- -1 , i ] x j - - I , j ] the open squares ( 3i - 2 , then from the squares [ -i 9 3 3 3 3 3i - 1 ) x ( 3j - 2 , 3j - 1 ) , and so on. Show that, for the set K remaining 9 9 9 from Q, [K, @2 I KJ is a continuum. 18. With the notations of exercise 7.1.f.15, suppose that every space [E;, 8!;] is a T2-continuum. Show that (a) If ii, . .. , in, j E/, i, ~ j (s = 1, ... , n) then A;1 , • • • , in;i equipped with the subspace topology obtained from X 8'; is homeomorphic with the space iE/

[Kx XE;, gri x iE/ i-:/=is,I

X

8fd

iEl

i¢is,i

where Kc E; x ... xE;. x~ is the image of the space ~ by the map given by the formula g(x) = (p 1jx), .. ,,P;.j(x), x) and 8f' = 8';1 x ... x 8';. x8fi I K; (b) The map g is (8fj, 8f')-continuous; (c) A;, ... ,; ·i is a continuum; (d) [E, 8f] is·~ T2-continuum. 19. Show that every asteroid [Ai, 8!1 ] is connected. [(10.1.11), (10.1.16).] 20. Show that every open star-countable cover of a connected space is countable. [Use the method of the proof of (8.3.58).] 1

414

10.2.

(10.2.1)

LOCALLY CONNECTED SPACES

21. Show that the asteroid [A1' 8f1 ] is not strongly paracompact if !is uncountable. (Observe that this space is metrizable, hence paracompact.) 00 22. In the space [R, @], let R =Au B, A # 0 # B, An B = 0, A = LJ Am 1

00

B =

U Bm

An and Bn closed for each n. Show that there is an open interval I

1

such that either I c A and one of the end-points of I belongs to B or I c Band one of the end-points belongs to A. [By (10.1.16), (10.1.9), and (2.2.20) mar A = mar B = C # 0; by (9.2.8) there is an open interval J such that J n C # 0 and either J n C c A,. or J n C c Bn for some n; say J n Cc A. By (10.1.9) and (10.1.16) the components of R - C are open intervals lying entirely either in A or in B, and J necessarily intersects one of the second type.]

10.2. LOCALLY CONNECTED SPACES 10.2.a. Local connectedness. The topological space [E, 8f] (or the topology 8f) is said to be locally connected if every point has a neighbourhood base consisting of connected sets. The following theorem contains further important characterizations: (I 0.2.1) Let [E, 8!] be a topological space. The following statements are equivalent: (a) The space is locally connected; (b) The components of every open set are open; (c) The space has a base consisting of connected sets. Proof. (a)= (b): If G is open and C is a component of G, then any point x EC has a connected neighbourhood V such that x E V c G; by (10.1.21) V c C, thus x Eint C and C is open. (b) = (c): The collection of all connected open sets constitutes a base since every open set can be written by (10.1.20) as the union of its components and thus as the union of connected open sets. (a): Obvious. I (c)

=

By using (10.1.16) we can see at once that each one of the following spaces is locally connected: [R, $], [[a, b], $ I [a, b]] (a, b ER, a~ b}, [[a, b), $ I [a, b)] (a < b). On the other hand the space [Q, &; I Q] is not locally connected as the components of the space itself are not open. The above space is not connected either. A well-known example of a connected but not locally connected space (moreover a continuum) is the following: let Ebe, in the plane, the union of the segment K = {(0, y) : - 1 ~ y ~ l} and the curve C = { (x, sin : ) : 0 < x

~ ~ } and consider the space [E, @ I E]. 2

It can be easily

seen that E is a bounded closed set in the plane; thus it is compact by (5.3.8) and connected. In fact, C is connected (it is the continuous image of the interva1

(10.2.4)

10.2.a.

415

LOCAL CONNECTEDNESS

(o, ~ Jso that (10.1.16) and (10.1.15) can be applied)andevidently E = C, thusE is connected by (10.1.10). The fact that this space is not locally connected can be seen from the fact that the components of a suitable open set are not open. Indeed, let G be the subset of

~

E lying in the open strip {(x, y): -

interval K' =

{ 0 such that every set having a diameter smaller than B is contained in it single element of 2!. Proof. For every x EE, look for a number Bx > 0 such that S(x, 2Bx) c A E m: n

for a suitable A. Then, by using {5.3.1), let E

=

LJ S(x;, Bx,) and

B = min(ex,, .. ,

1

exJ Then, for a set B C E, if b(B) < B, choose x E B n S(x;, Bx,) and then BC S(x, e) c S(xi, 2Bx) c A E 2-l:. I

By using this we can state: (10.4.3) Sierpinski's theorem. A pseudo-metric CO"(ltinuum [E, p] is .locally connected ifJ, for every B > Q, there is a cover of E consisting of a finite number of continua with diamrter less than e. Proof. The necessity results from (10.4.1) when applying it to the open cover

J. Conversely

{ S (x, ; ) : x EE

if the condition is fulfilled then choosing for

(10.4.4)

10.4.a.

the given open cover

Ki is a continuum and

m an

431

COVERING WITH CONTINUA

n

e > 0 according to (10.4.2), let E = U Ki }Vhere

J(Ki) < e. Clearly {Ki, ... , Kn}'~

1

m.1

10.4.b. Continuous images. We know from (10.1.15) that a continuous image of a connected space is connected as well; we can add to this by (5.3.10) that a continuous image of a continuum is a continµum as well. An important consequence of the preceding theorems is that, in S 2-spaces, the same can be said about locally connected continua:

(10.4.4) If [X, 8fi] and [Y, 8f2] are S 2-spaces,f: X---+ Y is a continuous surjection, and [X, 8f1 ] is a locally connected continuum, then [Y, 8f2 ] has the same property. Proof. Let mbe an open cover of Y; then {f- 1(A): A E m} will be an open n

LJ K 1 where Ki is a continuum and, for all i, there 1 n 1- 1(Ai). Hence Y = U f(Ki), f(Ki) is a continuum

cover of X and by (10.4.1) X = is an Ai E msuch that K 1 c

1

by (10.1.15) and (5.3.10), and further f(Ki) c Ai. The statement follows from (10.4.1).1 ' According to (10.4.4) every continuom; image of the space [I,~/ I] which is an S 2-space is a locally connected continuum. It is an important ~ and surprising - fact that this statement has a converse for metrizable spaces. The proof of this is based on a series of ~tatements which are interesting in themselves and deal with the representation as continuous images of the spaces of the given type. As a preliminary step, consider on the real line the Cantor set C, which'. is obtained by omitting from the interval I = [O, 1] the open interval (-}, ting from the remaining two closed intervals the intervals ( ~ ,

~)

!),

by omit-

and ( ; ,

!),

and .so on. We know that every element of I can be written in the form (*) w:here en takes the values 0, 1, 2 and the coefficients en are uniquely determineµ by xexcept for those xwhich can be written in the form

x = ~ ;:

I

m EN, cm ';/:- 0 (namely these can be written in another form x ·~ .

OC)

L 1

for some c,· ' 3: where

c~ = en if n = 1, ... , m - 1, c;.; = cm - I, and c~ = 2 if n > m) .. It can be easily seen that C consists of the points x E I· which can be written in the form I x =

i ;:

where en can be equal to O or 2 only as the open intervals omitted in

the nth step consist of those x for which in the representation (*) en is necessarily

10.4.

432

(10.4.5)

LOCALLY CONNECTED CONTINUA

equal to 1. It is known from (7.1.61) that [C, SI C] is homeomorphic with the space [PN, ®N] where P is a set consisting of two points and ® is the discrete topology of P. We shall show that every compact metrizable topological space is the continuous image of the space [C, @ I C]; first we shall show this for a special class of compact metrizable spaces. (10.4.5) The space [I, @ I I] is the continuous image of the space [C,@ I C].

f ;: , en = 0 or 2.- According to the above this

Proof. For x EC let x =

1

representation of x is unique; thus a map f: C . ing prescription: •

oo

Cn

1f x = ~ Evidently O ~ f(x)

3" (en =

~ ~ ~f

-> I can

be defined by the follow-

0, 2) then f(x) =

1

2

f 2" · oo

Cn

;n = I. S'ince every y El can be written in the form

1

y =

~

~: (dn = 0 or 1) (perhaps in two distinct ways) therefore if x =

~ ~~n

then x E C and f(x) = y hence f(C) = I. Further f is continuous, for if oo

Cn

1

~

then Ix - x'

I
m then either an = 0 or an = 2 (and accordingly either en = dn = 0 or en = dn = 2). Hence in such a case x is one (left or right) end-point of an open interval which has no common points with C but its end-points belong to C and one of y and z is the other end-point of this interval while the other has the property that between it and x there are further points of C. Define now the map f: C - F as follows: if there can be found, for the point x EC, a single point in F which is the nearest to it, letf(x) be this point; if there are two points with this property belonging to x, let f(x) be the one for which there is a further point of C between x and /(x). Evidently x E F implies f(x) = x; therefore f(C) = F. If x EC - F and there is a single point in F which is the nearest to x then the same point of F will be the nearest to every further point of C in a neighbourhood of x small enough; the same can be said if x = y; z, y, z E F, y < z but there is no point of Fin (y, z) as in this case, in a neighbourhood of x small enough, every point of C is on the same side of x as /(x). Therefore f is continuous at the points of C-F; and at the points belonging to F the same holds, since, for x in the neighbourhood of radius

~ of such a point x 0 E F, lf(x) - x I ~ I x 0

-f(x0)

I~

-

x I < ; , thus lf(x)-

lf(x) - x I + Ix - f(xo) I = lf(x) - x I + Ix -

Xo

I
0, there is a o > 0 such that, for O < p(x; y) < o, there exists in E an arc whose end-points are x and y and its diameter is smaller than 8. In fact E, equipped with the metric p, is complete by (5.2.19) and therefore 8f is, on account of (10.3.10) and (10.3.11), arcwise connected and locally arcwise connected. By using (10.3.5) an open cover of E can be given consisting of arcwise connected sets with diameter smaller than 8 and by (10.4.2) we can find to this a o > 0 such that, whenever O < p(x, y) < o, x and y can be included in one element of the cover and then they can be joined by an arc with a diameter smaller than 8. Since by (5.3.29), for o >0, there is an 17 >0 such that u, v EC, I u-v I < 17 implies p(f(u),f(v)) < o therefore finally there is, for 8 > 0, an 17 > 0 .such that u, v EC, Iu-v I< 17 implies either f(u) = f(v) or we can find an arc in E joining f(u) to f(v) with a diameter smaller than 8. Let now, for f(an) = f(bn) = x, g(u) = x in the whole interval [am bnl· On the other hand if f(an) =I= f(bn) then define g on the segment [am bn] in such a way that, with the notation g(fam bn]) =;:Am g I fa!,b.J be an ($ I [am bnJ, 8r I An)-homeomorphism, thus An be an arc in E joiningf(an) to f(bn), and o(AJ be smaller than the greatest lower bound multiplied by 2 of the diameters of the arcs in E with end-points f(an) and f(bn). According to our preliminary remark bn - an -+ 0 implies that o(An) -+ 0. g: I -+ E defined in this way is indeed continuous. This is evident at a point u0 E (am bn). If u0 E C and u0 is the end-point of an interval (am, bm), say u0 = bm, then there is first, to 8 > 0, a o1 > 0 such that u0 - 01 E · E (am, bm) and u0 ~ o1 < u ~ u0 implies p(g(u), u(tto)) < 8. Then there is a 02 > 0 such that u0 that u, v EC,

~

+ o2, u EC implies p(f(u),f(tto)) < -~, and a 03 > 0 such I u -v I < o3 implies either f(u) = f(v) or thatf(u) andf(v) can be u< u0

10.4.c.

(10.4.9)

435

EXERCISES

8

joined in E with an arc having a diameter smaller than 4 . If o4 is the diameter




compact, hence

fl Wn(x)

consists of exactly one point;

1

00

(g) XE I implies; with the notation {J(x)}

= n Wn(x),

!(int Vn(x)) C Wn(x)

1

and hence /: I ~ Q is (@ I 1, @2 I Q)-continuous; (h) If y EQ, we can find numbers k 1 2 , ••• (kn = 1, ... , 9) such that y E

,k

00

E Qk, ...kn for every n and then

nIk, ...kn consists of exactly 1

· which Qk, ...kn C Wn(x) (n E N); (i) f is· surjective.

one point X E I for

11. TOPOLOGICAL GROUPS

ll.L GROUPS 11.1.a. The notion of group. It often happens that on a topological space some algebraic structure is defined comprising one or more operations, e.g. on the set R the addition and multiplication of real numbers. The theory of such topological spaces equipped with an algebraic structure is an important meeting ground of topology and algebra. In the following we shall study only one such aspect, namely the theory of topological groups (and only its elements). We understand by an operation of m variables on the set E =I= 0 a map of the set Em into the set E (e.g. the addition and multiplication of real numbers are operations of two variables on the set R). The set Eis said to be a group if there is an operation of two variables n: E. x E -+ E defined on E which is associative. That is, x, y, z EE implies

ft(n(x, y), z) = n(x, n(y, z)),

further there is in E an element neutral with respect to n, i.e. an element e EE such that x EE implies · n(x, e) = n(e, x) = x,

and finally every x EE has an inverse with respect to n, i.e. a map t: E-+ E can · be defined such that if x E E then n(x, t(x)) = n(t(x), x) = e. It is evident from the. definition that the group is defined by giving the set E and the operation n, hence we should call the group the couple [E, n]; however we shall not need this more precise notation since we shall not simultaneously · · define on the same set groups with different operations. Instead of the foregoing complicated notation the multiplicative notation of groups is more usual which consists of denoting the map n by writing the variables one beside the other -as we do for multiplication and denoting the inverse of x by x- 1 . The above conditions can be written by means of this not!!,tion more concisely as follows: · (CJ (xy)z = x(yz) (x, y, z EE); (C2) There is an e EE such that xe = ex· = x (x EE); (C3) For every x EE, there is an x.,. 1 EE such that xx'.'"" 1 = ·x-:1,x == e.

438

11.1.

(11.1.1)

GROUPS

The group is commutative or Abelian if moreover (C4) xy = yx (x, y EE). The additive notation is also used in commutative groups; the map n is denoted then by + on the pattern of the addition and the inverse of x by (-x):

n(x, y) = x

+ y,

t(x) = -x.

The neutral element is also called the unit element in the multiplicative notation and the zero element in the additive one. Further usual simplifying notations are: in the case of multiplicative notation

(xy)z

= x(yz) = xyz,

xx = x2, x2x = x3, ... ,

and in the case of additive notation

+ x = 2x, 2x + x = 3x, ... , -2x, (-2x) + (-x) = -3x, ... , x + (-y) == x -y. x

(-x)

+ (-x) =

We spoke above of "the" neutral element; this is motivated by: (11.1.1) If x, y EE implies xy EE, (C1) is fulfilled and (CJ holds for e1 and e2 instead ·of e, then e1 = e2 • Proof. Applying (CJ for x = ei, e = e2, then for x = e2, e = ei, it follows e1e2 = e2e1 = e1 , e2e1 = e1e2 == e2, i.e. e1 = e2• I An analogous remark can be made in connection. with (C3): (11.1.2) If x, y EE implies xy EE, (CJ and (C2) are fulfilled, further x 1 x = = xx2 = x{x = xx~ = e, then x1 = x{ = x 2 = x~. Proof. x1 = x 1e = x 1(xxJ = (x1 x)x2 = ex2 = x 2 ; similarly x{ = x~ and x{ ;= = X2-I 11.1.b. Examples. (a) If E = R, n(x, y) = x + y, then -we obtain a group, e = 0, t(x) = -x (tlµs is the origin of the additive notation and terminology). (b) E = Q, n and ·t as before. (c) E is the set of integers, .n and (d) E

= R - {O}, •

n(x, y) = xy, e

t

as before.

~

I, i(x)

= __!_ X

multiplicative notation and terminology). (e) E = Q - {O}, n and t .as before. {f) E = {x: x ER, x> O}, n and i .a,.s before.

(this is the origin of the

(11.1.3)

11.1.b,

EXAMPLES

439

(g) E = Rm, n(x, y) = z, where x = (xl, ... , Xm), y = (Yi, ... , Ym), z = = (z1, .•• , zm), Z; = X; + Yi (i = I, ... , m), further e = (0, ... , 0), t(x) = = (-Xi, ... , -xm). (h) Eis the set of the real functions defined on the set H =f,. 0, n(x, y) = x + y, e is the function identically 0, t(x) = -x. (i) Eis the set of the complex numbers, n(x, y) = x + y, e = 0, t(x) = -x, (j) Eis the set of the complex numbers distinct from 0, n(x, y) = xy, e = I, 1 t(x) = - . .

X

(k) E is the set of the complex numbers with the absolute value 1, n and t as before. (1) E is the set of the bijective maps of the set H =f,. 0 onto itself, n(f, g) = =Jo g, e is the identity map of H, t(f) = 1- 1. Now (C1) is a consequence of (2.6.6).

Groups (a)-(k) are commutative, in general group (I) is not. Let H = R, and f(x) = x + a, g(x) = 2b - x (thus J is the translation of R by a while g is the reflexion of R with respect to b where a, b ER are given). Now, with the notation h = J o g, k = g o J, h(x) = 2b

+a

- x, k(x) = 2b - a - x.

Hence a =fa O implies f o g =fa g o f In the following the multiplicative notation will always be used (with the exception of some concrete examples). 11.1.c. Multiplication of sets. The following notation will be very useful: If A and B are two subsets of the group E, then AB= {x: x = ab, a EA, b EB},

A- 1

=

{x:x

=

a- 1 , a EA}.

It follows at once from the definitions that: (11.1.3) If E is a group, A, B, C, A 1 , B 1 c E, then (a) AB = 0 if! A = 0 or B = 0; (b) A- 1 = 0 if! A = 0; (c) A(BC) = (AB)C, EE = E; (d) (AB)- 1 = B- 1A-1; (e) (A- 1)- 1 = A; (f) A c A 1, B c B1 imply ABC A 1Bv· (g) A c A 1 implies A- 1 C A 1 1. Proof. We have to note that

AB= n(A x B), A- 1 = t(A),

11.1.

440

GROUPS

(11.1.4)

further for (d) that x, y EE implies (xy)- 1 = y- 1x- 1 since (xy)(y- 1x- 1) = x(yy- 1)x- 1 = xex- 1 = xx- 1 = e, (y-lx-l)(xy) = y-lcx-lx)y = y-ley = y-ly = e, while for (e) that (x- 1)- 1 = x.

I

If A = {a} has a single element then we write aB instead of {a} B and Ba instead of B{a}.

11.1.d. Translations. If Eis a group and a EE, then certain maps of E onto itself are defined by the formulae

1:!(x) = ax, 1::(x) = xa. 1:; and 1:: are respectively the left translation and the right translation associated with a. (11.1.4) For any group E and element a EE, 1:;: E - E and 1::: E - E are bijective, if a, b EE then

and Proof. x EE implies

(1:~

o

1:b)(x) = 1:!(1:t(x)) = a(bx) = (ab)x = r!ix),

(1::

o

1:i)(x) = 1::(1:i(x)) = (xb)a = x(ba) = 1:iaCx).

Thus 1:~ 0 1::-1 = '1:~-1 0 1:: = 1:~, and

r! =

1:: is the identity map of E.

I

11.1.e. Subgroups. If E is a group, 0 :/= E 0 c E, and the set E 0 constitutes a group with the operation n I E0 x E 0 , then we call E0 a subgroup of the group E.

(11.1.5) Let Ebe a group, 0 :/= E0 c E. The following statements are equivalent: (a) E 0 is a subgroup of E; (b) x, y E E0 implies xy EE0 , x- 1 E E0 ; (c) x, y EE0 implies xy- 1 EE0 ; (d) x, y EE0 implies x- 1y EE0 • Proof. (a) => (b): If x, y EE0 , then necessarily xy EE0 • If e0 is the neutral element of E 0 then x EE0 implies xe0 = x, thus x- 1(xe0 ) = x- 1x, i.e. ee0 = e, e0 = e and e EE 0 • If x 1 is the inverse of x EE 0 in E 0 , then xx1 = e, thus x- 1(xxJ = = x- 1, ex1 = x- 1 , x 1 = x-1, and x- 1 E E0 •

(11.1. 7)

11.1.d.

TRANSLATIONS

441

(b) ~ (c), (b) ~ (d): Obvious. (c) ~ (b): xx-I= e implies eEE0 , further xEE0 implies ex-I= x-IEE0 , finally x, y E E0 implies xy = x(y- 1 )-I E E0 • (d) ~ (b): In the same way. (b) ~ (a): x E E0 implies xx-I = e E E0 • (CI), (C 2), and (C 3) are evidently fulfilled for E0 with the same neutral element and inverse as in E. I Among the examples under 11.1.b (b) and (c) are subgroups of (a), (e) and (f) are subgroups of (d), (k) is a subgroup of (j). If [H, 8T] is a topological space, then the homeomorphisms of the space onto itself constitute a subgroup of the group (I) namely (11.1.5) (b) is fulfilled as the composition of homeomorphisms and the inverse of a homeomorphism are homeomorphisms as well. If E0 is a subgroup of the group E, then the sets having the form aE0 or E0 a are called the left cosets or the right cosets with respect to E0 • (11.1.6) Let E be a group, E 0 a subgroup of E, a, b EE. aE0 n bE0 ¥- 0 implies aE0 = bE0 • The left cosets constitute a partition of E. Proof. If c = aE0 n bE0 , say c = ax = by, x, y E E 0 , then, for arbitrary z E E0 ,

az = axx- 1z = b(yx-Iz) EbE0 , and thus aE0 c bE0 = ae EaE0 • I

c aE0 • The second statement follows from this in view of a =

Of course analogous statements are valid for the right cosets too; in general, in every definition or statement concerning groups, the roles of multiplication on the left and on the right can be interchanged. 11.1.f. Homomorphisms. The map h: E-+ E' from the group E into the group E' is said to be a homomorphism if x, y EE implies h(x)h(y) = h(xy).

(11.1. 7) If E and E' are groups, h: E -+ E' is a homomorphism, then h(E) is a subgroup of E', h(e) is the neutral element of E', and h(x- 1) = h(x)- 1 • Proof. x, y EE implies h(x)h(y) = h(xy) Eh(E), further if x EE then by

h(x)h(e) = h(e)h(x) = h(xe) = h(ex) = h(x) h(e) is a neutral. element in h(E), and finally by h(x)h(x-I) = h(x-I)h(x) = h(xx-I) = h(x- 1x) = h(e) h(x- 1) is the inverse element of h(x) in h(E). Hence h(E) is a subgroup of E'. It follows from (11.1.5), (11.1.1 ), and (11.1.2) that the neutral element of h(E) is the same as that of E' and the inverse in h(E) of an element of h(E) is the same as the one in E'. I

442

11.1.

GROUPS

(11.1.8)

(11. I .8) If E and E' are groups, h: E ---+ E' a homomorphism, then A, B c E implies h(AB) = h(A)h(B), h(A)- 1 = h(A- 1). I (11.1.9) Let E and E' be groups, h: E ---+ E' a homomorphism, e' the neutral element of E'. Then E0 = h- 1 (e') is a subgroup of E for which the. left cosets and the right cosets coincide. Proof. x, y EE, h(x) = h(y) = e' imply h(xy) = h(x)h(y) = e' e' = e', further h(x- 1) = h(x)- 1 = e'- 1 = e'. Hence h-1(e') is a subgroup by (11.1.5). If a EE, x E E0 , then by

h(axa- 1) = h(a)h(x)h(a- 1) = h(a)e'h(a)- 1 = e' y = axa- 1 E E 0 thus ax = ya and aE0 c E0 a. By means of an analogous reasoning a- 1xa E E0 and E0a C aE0 .1 The subgroups occurring here, whose left and right cosets coincide, are very important; we call them normal subgroups. In a commutative group every subgroup has this property. (11.1.10) The subgroup E0 of the group E is a normal subgroup if! a EE, x E E0 implies axa- 1 E£ 0 • Proof. If the left and the right cosets coincide, then by (11.1.6) a EE implies aE0 = E 0 a, i.e. x E E0 implies ax = ya, axa- 1 = y E E0 for a suitable y E E 0 • Conversely if a EE, x E E0 imply axa- 1 E E0 , then aE0 a- 1 c E0 , i.e. aE0 a- 1a c E0 a, aE0e C E0 a, aE0 C E0 a and in an analogous way a- 1E0 a C E0 implies E0 a C aE0 • I

The following converse of (I 1.1.9) shows the importance of normal subgroups: (11.1.11) Let Ebe a group, E0 a normal subgroup ofE, E' the set of cosets belonging to E0 • Then E', with the operation x'y' in 11.1.c, is a group, its neutral element is e' = E0 and. the surjection h: E ---+ E' mapping x EE to the coset containing it is a homomorphism for which h- 1 (e') = E 0 . Proof. If x Ex' EE', y Ey' EE', then by (11.1.10) x' = xE0 = E0 x, y' = = yE0 = E0 y, thus x'y' = xEoE0 y = xE0 y = xyE0 EE', further Eo(xE0 ) = xE0E0 = = xE0 , (xE0 )E0 = xE0 so that e' = E0 is indeed a neutral element, finally xE0 E0 x- 1 = xE0 x- 1 = E0 , E0 x- 1xE0 = E 0 shows that E0 x- 1 is the inverse of xE0 • Accordingly E' is in fact a group. Since x Ex' EE', y E y' EE' imply xy E x'y', therefore h is indeed a homomorphism. I

The group E' constructed in (11.1.11) is called the factor group of E by £ 0, and is denoted by E/E0 • If E and E' are groups, h: E ---+ E' is a homomorphism and h is injective or surjective, then it is called a monomorpbism or an epimorphism; if h fulfils both conditions simultaneously (i.e. if it is bijective) then it is called an isomorphism. It follows at once from the definitions by (2.6.5) that: (11.1.12) If h:E---+E' and h':E' ---+E" are homomorphisms, monomorphisms, epimorphisms, or isomorphisms, then so is h' o h. I

11.1.g.

(11.1.13)

443

EXERCISES

(11.1.13) If h: E ~ E' is an isomorphism, then so is h- 1 . Proof. Clearly h- 1 is bijective. However, x', y' EE', x = h- 1(x'), y = h- 1(y') imply h(xy) = h(x)h(y) = x'y', i.e. xy = h-1(x'y') and h- 1 is a homomorphism. I

The groups E and E' are said to be isomorphic if there exists an isomorphism h: E ~ E'. (11.1.12) and (11.1.13) show that this is an equivalence relation. 11.1.g. Exercises. 1. The set Eis said to be a (real) vector space if an operation n:E x E~E denoted by n(x,y) = x + y is defined in E and also a mapµ: R x E ~ E denoted by µ(a, x) = ax is defined such that x, y, z EE, a, PER imply

x+y=y+~x+(y+~=~+~+~ a(Px) = (aP)x, (a a(x

+ y) =

ax

+ P)x

+ ay,

= ax

+ Px,

Ox = Oy, lx = x.

Show that E is then a commutative group with the operation n. [Its zero element is that o EE for which o = Ox for every x EE, and the inverse of x is (- l)x.] 2. Show that the following sets, with the given maps n and µ, constitute vector spaces: (a) E = Rm, (x1, ... , Xm) + (Yi, ... , Ym) = (x1 + Yi, ... , Xm + Ym), r:t.(Xi, ..• , Xm) = (r:t.X1, ... 'IY.Xm);

(b) Eis the set of all real functions defined on the set H =I= 0, n(x, y) = z if z(t) = x(t) + y(t), µ(a, x) = y if y(t) = ax(t) (t EH); (c) Eis the set of continuous functions defined in the topological space [H, ST], n and µ are as before; (d) E is the set of all bounded proximally continuous functions defined in the proximity space [H, Si'], n and µareas before; (e) Eis the set of all uniformly continuous functions defined in the uniform space [H, "Ll'.], n and µ are as before. 3. Let E be the set of the bijections of a set H =I= 0 onto itself. If f, g EE then let n(f, g) = f o g. Show that the following sets E0 are subgroups of the group E: (a) E0 consists of those JEE which map a given element t0 of H to itself; ·(b) E0 consists of those f which are (Si', Sl')-equimorphisms with respect to a proximity Si' on H; (c) E 0 consists of those f which are unimorphisms with respect to a uniformity "Ll'. on H; (d) H c R is an interval and. E 0 consists of those f which are strictly mono· tone increasing and ($ I H, $ I H)-continuous; (e) His a group and E0 consists of those f which are isomorphisms. 4. In the group E under 11.1.b (a) define two sets E1 c E and E 2 c E which are not subgroups although (a) x, y E E 1 implies x + y E E 1 ;

444

11.1.

GROUPS

(b) x E E 2 implies -x E E 2• [E1 = (0, + oo), E 2 = (-1, 1).] 5. In the previous group, define a set Ea c .E which is a group with a suitable operation n however it is not a subgroup of E. [Ea= E ~ {O}, n(x, y) = xy.] 6. Let E be a group, E' the group of the bijections of E onto itself defined in 11.1.b 0) and let/: E-+ E' be defined by the formula f(x) = t!, Show that f is a monomorphism. 7. Let E be a group, and if a EE and x EE, then f(x) = axa- 1• Show that f: E -+ E is an isomorphism. 8. For a, b ER, a =I 0, let fa,b: R -+ R be defined by the formula fa.ix) = = ax + b. Show that (a) E = {fa,b: a ER - {O}, b ER} is a subgroup of the group, defined in 11.1.b (1), consisting of the bijections of the set R onto itself; (b) E 0 = Ua,b: a E (0, +co), b ER}, E 1 = {h,b : b ER}, and E 2 = {/a, 0 : a ER - {O}} are subgroups of E; (c) E 0 and E 1 are normal subgroups of E however E 2 is not a normal subgroup. [fa,b O fc,d O fa~l = fc,ad-bc+b·] 19. Let Ebe the group considered in the preceding exercise, h: E-+ R - {O} defined by the formula h(fa,b) = a. Show that, in the setE' = R - {O}, n(x, y) = = xy is an epimorphism from E onto E' and the corresponding normal subgroup is E1 . 10. Let E be a group, and E 0 a subgroup of E. Show that a, b E E belong to the same left (right) coset with respect to E0 iff a- 1b E E0 (ba- 1 E E0). 11. Show that the factor group by £ 0 of the group considered in exercise 8 consists of two elements. 12. Let Ebe the group, defined in 11.1.b (]), consisting of the bijections of the set R onto itself, let E 0 c E consist of those continuous strictly monotone functions/ for which/(R) = R, and let E1 c E 0 consist, of those/ which are strictly monotone increasing,. E 2 c E0 of those whose derivative exists and vanishes nowhere. Show that (a) E0 , E1 , E 2 a:i;e subgroups of E; (b) E 1 is a normal subgroup of E 0 and Eo/E1 consists of two elements; (c) E 2 is not a n.ormal subgroup of E0 ; . (d) E1 is not a normal subgroup of E. [If f(x) = x + 1, g(x) = x for x ~ 0, g(x) = 2x for xi;;; 0, then f E e2, g EEo and with the notation h =go f o g-1, h(x) = x + 1 if x ~ -1, h(x) = 2x + 2 if -1 ·~ x ~ 0, h(x) = x + 2 if x i;;; 0, thus h ~ E 2 • For the same f and for k ' ' ' 1 ' ' ' ' . defined by the formulae. k(:x) = - if ,x =I 0, k(O) = 0, we have f E Ei, k EE and, .

with the notation m

X

=k

m(-1) = 0, thus m~E1 .]

of o k-1, m(x)

,

X

= -,--. if x x+l

.

=/,

0, -1, m(O) = 1,

_ 11.1.g.

445

EXERCISES

n

13; - Let Ebe a group, Ei (i E /) a subgroup of E. Show that E; is a subgroup of E as well. iEI 14. Let Ebe a group, A c E. Show that the smallest subgroup of E containing A consists of all elements having the form b1b2 ••• bn where n E N and bi E B = = Au A- 1 (i = l, ... , n). 15'. Let I# 0 be an arbitrary set of indices, E; a group for i EI, E = X Ei. Show that · iEI (a) E is a group with the notation n(x, y) = (X;Yi) where x = (x;), y = (y;); (b) The set E* c E which consists of those elements x = (x;) EE for which xi = e; holds except for a finite number of indices (e; denotes here the neutral element of E;) is a subgroup of E; (c) The projection P; from E onto Ei, further Pi IE* are epimorphisms.

11.2. TOPOLOGICAL GROUPS 11.2.a. The notion of topological gr~up. We speak_ of a topological group if there is given, on a group E, a topology 8T" such that the maps n and ,_ are continuous with respect to 8T", i.e. more exactly n: E x E ---+ E is (&f x 8T", 8i)-, and t: E ---+ E is (8T", gf")-continuous. The topologies given on the group E and fulfilling these conditions are called the compatible topologies of the group. In example 11,1.b (a) $, in (b) $ Q, in (d) $ J(R - {O}), in (e) $ (Q - {O}), in (f) $ I E, in (g) t§m, in (h) the topology of th(? pointwise convergence, in (i) by identifying as usual the complex numbers with the points of the plane - _$ 2, in U) $ 2 I E, in (k) $ 2 I E are clearly compatible topologies. Hence all these are examples of commutative topological groups. It is also evident that any group equipped with a discrete or indiscrete topology is a,, topological group (since a map f: E ---+ E' is always continuous if the topology of E is discrete or that of E' is indiscrete). A less trivial example of a non-commutative topological group is the following: consider the bijections of the set R onto itself given by f(x) = ax + b (a > 0). These constitute a -non-commutative subgroup of the group consisting of all bijections of R, for if g(x) = ex + d (c > 0), then with the notations f o g = h, 1- 1 = k clearly · · 1 b · h(x) = acx + (ad+ b), k(x) = ---:-X - - • a a In other words, if fisidentified with the point (a, b) of R 2, then E = {(a, b): (a,b)ER2, a> O}, and if(a,b),(c,d)EE, then J

J

(a, b) · (c, d) = (ac, ad+ b), .

(a, b)

-1

a'_- ab } .

= (1

From this it can be seen at once that $ 2 I E is a compatible topology for E.

11.2.

446

TOPOLOGICAL GROUPS

(11.2.1)

ll.2.b. Neighbourhood bases of e. The topology of a topological group is entirely defined as soon as a neighbourhood base of the neutral element e is known. In order to prove this we notice first that: (11.2.1) In a topological group any left or right translation and t are homeomorphisms of the group onto itself. Proof. If [E, 8f] is a topological group, a EE, then ,! = n o f, where f: E-+ Ex Eis the map defined bytheformulaf(x) = (a, x). Thelatteris (gf, 8l x 8f)continuous by (7.1.28), hence ,! is (8f, 8f)-continuous and t]:ie same holds of course for '~-' i.e. by (I 1.1.4) for (,!)- 1 as well. The statement for ,~ can be proved in a similar way. Finally tis bijective, (8f,gf)-continuous and by i- 1 =1, a (8f,8f)-homeomorphism. I

We obtain at once from this by the evident formulae ,!(e) = ,~(e) = a, ,!(V) = aV, ,~(V) = Va: (11.2.2) If in the topological group [E, 8f] b is a neighbourhood base of the neutral element e, then both {a V: VE b} and {Va: VE b} are neighbourhood bases of the element a EE. I

The question arises quite naturally what conditions has the system of sets b consisting of subsets of the group E to fulfil in order that it be a neighbourhood base of e with respect to a compatible topology. Concerning this point we can say: (11.2.3) If b is a neighbourhood base of e in the topological group [E, 8f], then (a) VE b implies e E V; (b) b is a filter base; (c) if VE b then there is a Vi E b such that Vi Vi C V; (d) if VE b then there is a V1 Eb such that v1 - 1 c V; (e) if VE b, a EE, then there is a Vi E b such that a V1a- 1 c V.

Proof. (a) and (b) are evident. (c): In view of n(e, e) = e there are, for VE b, V', V" Eb such that n(V' x x V") c V, and if V1 Eb is such that Vic V' n V", then n(Vi x V1) c V, i.e. V1 Vi c V. (d): In view of t(e) = e there is, for VE b, a Vi Eb such that t(Vi) CV, i.e. v1- 1 CV. (e): By (11.2.2), for VE b and a given a EE, there is a Vi E b such that aVi c c Va, i.e. aVia- 1 c Vaa- 1 = Ve= V. I Now the conditions of (11.2.3) are also sufficient. This is the consequence of the following theorem which asserts even more than this: (11.2.4) Let the system of sets b consisting of subsets of the group E fulfil conditions (a)-(e) in (11.2.3) and define, for VE b,

Uv = {(x, y): x- y E Vn v1

ut =

{(x, y):

xy- 1

E Vn v-

1}

1}

c Ex E, c Ex E.

(11.2.5)

11.2.b.

NEIGHBOURHOOD BASES OF e

447

Then us = { Uv: VE b} and ua = { Uf: V E b} are uniform bases for uniformities "Us and 61£d respectively on E, 8!'LI• = 8T'Lid is a compatible topology and b is a neighbourhood base of e, while {aV: VE b} and {Va: VE b} are neighbourhood bases of a EE with respect to this topology. Proof. If x EE then by (a) x- 1x = e E V n v- 1 , thus (x, x) E Uv. As V1 c V2 implies V1 n v1- 1 c V2 n v2-1, therefore Uv, c Uv,, and from this we obtain

by (b) that U' is a filter base. If (x, y) E Ut,, i.e. if x- 1y E V n v-1, then y- 1x = = (x- 1y)- 1 E v- 1 n (v- 1)- 1 = v- 1 n V, in view of (I 1.1.3); accordingly Uv is a surrounding. If ViVi c V, then (x, y) E Uv,, (y, z) E Uv, implies x- 1y E Vin n v 1- 1, y- 1z E Vi n v 1- 1, thus

x- 1z = x- 1yy- 1z E V1V1 n

c

v1- 1 v1-l C

V n v-1,

us

so that Ui,, o Uv, Uv. Accordingly is a uniform base for a uniformity "Us. The same can be stated for Ud and 6Ua. The sets Uv(a) or Uf(a) (VE b) constitute a 8f'LI,- or 8T'Lld-neighbourhood base respectively of the point a EE. However clearly

v- 1), (V n v- 1)a,

Uv(a) = a(V n Uf(a) =

· and it follows from (d) that b1 = {Vn v- 1 : VE b} is a filter base equivalent to b since v1- 1 c V, V2 c V n Vi implies V2 c V n v- 1 , so that {aV: VE b} and {Va: VE b} are equivalent, respectively, to the two neighbourhood bases in question. Since by (e) there are, for VE b, a V1 Eb such that aV1 c Va, and a V2 E b for which a- 1 V 2a c V, V2a c a V, therefore ~he two neighbourhood bases are equivalent to each other. Hence 8f'LI, = 8!'Lld and, denoting this topology by 8T, if a = e we obtain that b is a ST-neighbourhood base of e. If a, b, c EE, ab = c, and Vi Eb is chosen for VE b according to (c) in such a way that Vi Vi c V, further V2 E b is such that V2b c b Vi, then a V2 and b V1 are 8!-neighbourhoods of a and b respectively and n(a V2 x b Vi) = a V 2b Vic ab V1 Vi c c c V so that n is (8T x 8!, 8!)-continuous. On the other hand, if Vi E b is chosen for VE b according to (d) such that v1- 1 C V, then, by t(a11 V1 ) = (a- 1 V1 )- 1 = = v1- 1a c Va, t is (8T, 8!)-continuous and § is compatible. I The definition of the surroundings Uv and Uf will become simpler if V is symmetric, i.e. if v- 1 = V; if the filter base b consists of symmetric sets, it is itself called symmetric. In connection with this we notice that: (11.2.5) In any topological group e has a symmetric neighbourhood base. Proof. It can be seen from condition (11.2.3) (d) that if b is an arbitrary neighbourhood base of e then { V n v- 1 : VE b} is a symmetric neighbourhood base of e. I We can state an important property of the uniformities 6l£s and 6l£d in (11.2.4). For this purpose, let us call, in the group E, the surrounding U left or right in-

11.2.

448

(11.2.6)

TOPOLOGICAL GROUPS

variant if a EE, (x, y) E U implies (ax, ay) E U or (xa, ya) E U respectively, while the uniformity 6lf on Eis said to be left or right invariant if it has a base consisting

of left or right invariant surroundings. Now: (11.2 . 6) The surrounding Uv in (11.2.4) and hence the uniformity q,es are left invariant, similarly U# and 61,fd are right invariant. Proof. a EE, (x, y) E Uv imply (ax)- 1ay = x- 1a- 1ay = x- 1ey = x- 1y E V n n v- 1, thus (ax, ay) E Uv, Similarly (x, y) E ut implies (xa, ya) E ut,

I

The following fundamental theorem is now easily obtained from our results: (11.2. 7) Let [E, ST] be a topological group. Then there exist on Ea unique left invariant and a unique right invariant uniformity inducing ST; these are identical to the uniformities q,es and 61,fd respectively constructed in (11.2.4) starting from an arbitrary ST-neighbourhood base b of e and are called the left and right uniformities of the group. Proof. Let b be a SJ-neighbourhood base of e, q,es and 61,fd the two uniformities constructed in (11.2.4). By (11.2.6) q,es is left, 61,fd is right invariant and we know that {a V: VE b} and {Va: VE b} are neighbourhood bases of a EE with respect to ST'l.fs = ST'l.fd· Hence by (11.2.2) ST'l.fs = ST'l.fd = ST. If 6lf 1 and 6lf 2 are left invariant uniformities on E and ST'l.f, = ST'U, = ST, then, for any surrounding U2 E 6lf 2 , there exists a left invariant surrounding U~ E '\f 2 such that Vi c U2• As U~(e) is a SJ-neighbourhood of e, there exists a left invariant surrounding U1 E '\f 1 such that Ui(e) c U~(e). Now if (x, y) E U1 then (x- 1x, x- 1y) = (e, x- 1y) E Ui, i.e. x- 1 y E Ui(e) c U~(e), thus (e, X- 1y) E U~, (x, xx- 1y) = = (x, y) E Vic U2 so that U1 c U2 and '\f 2 < 6lf 1 . In' the same way '\f1 < '\f 2, thus '\f 1 = '\f2 . The statement for right invariant uniformities can be proved in the same way.

I

Condition (e) in (11.2.3) is automatically fulfilled in the case of a commutative group E, namely Vi = V can be chosen for an arbitrary a EE; in this case the left invariant and the right invariant uniformities coincide with each other and q,es = = 61,fd, However this equality can also hold in the case of a non-commutative group; e.g. if SJ is the discrete topology, then qes and 61,fd are clearly identical with the discrete uniformity of E whether the group Eis commutative or not. Similarly if SJ is compact then SJ is induced by a unique uniformity by (5.3.26), thus again

qes =

61,fd_

However it can occur that qes and 61,fd do not coincide. For example, let [E, ST] be the group mentioned at the end of the preceding section, i.e. E = {(a, b): (a, b) E R 2 , a> O}, (a, b) · (c, d) = (ac, ad+ b), ST = @2 / E. Then e = (I, 0) and the sets v. = (I - a, 1 + a)x (- a, a) (0 < a < 1) constitute a neighbourhood base of e so that Ut consists of those pairs ((a, b), (c, d)) for which C d-b). (a, b)- 1(c, d) = ( ~, - a - E v.

(11.2.9)

11.2.c.

449

CONSEQUENCES

and simultaneously

b- d)

a (c, d)- 1(a, b) = (--;; , - c - E V,, i.e. I

db I 0 is arbitrary, c = a, and I b - dI=

e a; at the same time the second and the fourth of the latter four do

not hold, e.g. for e =

~

as soon as ea > l. Accordingly

ut,,,

does not contain

any U~. at all. 11.2.c. Consequences. The fact that the topology of a topological group can be induced by a left or right invariant uniformity has important consequences. First of all:

(11.2.8) The topology of any topological group is completely regular.

I

(11.2.9) Let [E, 8T] be a topological group. The following statements are equivalent: (a) 8T is a T0-topology; (b) 3J is Tykhonov; (c) 611 8 is separated; (d) 61,£d is separated; (e) e has a BT-neighbourhood base tJ such that n {V: VE tJ} = {e }. Proof. (a)~ (b) results from (11.2.8), (b) ~ (c) and (b) ~ (d) follow from (3.2.24). 29 Akos Csaszar

11.2.

450.

TOPOLOGICAL GROUPS

(11.2.10)

(c) = (e): 8f is a T,,- and Ti-topology, so that, if x EE - {e}, there is a VE tJ such that x 4 V where tJ can be an arbitrary neighbourhood base of e. (d) = (e): In the same way. (e) = (a): If x, y EE, x f= y, then there is a VE tJ such that x- 1y 4 V, i.e. y 4 x V. We obtain from this by (11.2.2) that 8f is a T0-topology. I A topological group fulfilling condition (11.2.9) (e) is said to be separated. A pseudo-metric or metric p defined on the group E is said to be left or right invariant if x, y, a E E implies p(ax, ay) = p(x, y), or p(xa, ya) = p(x, y), respectively. The following important theorem can now be proved: (11.2.10) Let [E, 8!] be a ( separated) topological group. The following statements are equivalent: (a) 8f is pseudo-metrizable (metrizable); (b) e has a countable neighbourhood base; (c) 8f can be induced by a left invariant pseudo-metric (metric); (d) 8f can be induced by a right invariant pseudo-metric (metric). Proof. (a)= (b) results from (2.4.11). (b) =i, (c): Let { Vn: n = 0, I, 2, ... } = b be a countable neighbourhood base of e. By using (11.2.3), starting from V~ = Vo, let - if V~ E bis defined already V~+ 1 Eb be such that V~+ 1 V~+ 1 c V~ n Vn+l (n = 0, 1, 2, ... ). Evidently b' = = { V~: n = 0, I, 2, ... } is a neighbourhood base of e. Therefore by (11.2.4) and (11.2. 7), with the notatioi:i Un = Uv;,, {Un: n = 0, I, 2, ... } is a uniform base generating 6Us and each Un is left invariant by (11.2.6), further Un+l o Un+l C Un. Therefore (4.2.29) can be applied according to which 6Us is induced by the pseudometric

0 with respect to the uniform continuity of the map g and, for n ~ n0, IfnCx) - f(x) I < b, Ifn-1(x) - g(x) I < e for every x E I, then for such n

it

I g(f(x))

- x

Therefore g(j(x))

I ~ I g(f(x))

- g(fn(x))

I + I g(fn(x))

-

1n- 1(fn(x)) I < 2e.

= x so that f is injective, i.e. strictly monotone and f EE.

(11.3.11) If 611b is the bilateral uniformity of a topological group, then i is (611b, 611b)-uniformly continuous. Proof. By (11.3.3) and (3.2.53) i is (611b, 611')- and (611\ 611d)-uniformly continuous and hence it is (611b, 611b)-uniformly continuous in view of(3.2.54). I ' . . ' , . '

'

In connection with a part of the reasoning in the previous example,, we can say:

(11.3.15)

11.3.d.

THE COMPLETION OF A TOPOLOGICAL GROUP

463

(11.3.12) If the topology gr of the topological group [E, gr] is (pseudo-) metrizable, then so are 61£•, 61,fd, and 61,fb. Proof. By (11.2.10) 8f = grPi = grP, where p1 is a left, p 2 a right invariant pseudometric. Then 6UP1 is left invariant and 'UP, is right invariant, thus by (1L2.7) 6UP1 = 61£•, 'UP, = 61£d. (4.2.34) shows that 61,fb is pseudo-metrizable as well.

I

(11.3.13) If E 0 is a subgroup of the topological group [E, gr] and the bilateral uniformity of the latter is 'Uh, then 61,fb E 0 is the bilateral uniformity of the topological group [Eo, gr I Eo]. Proof. It is a consequence of (11.2.11) and (3.2.37). J

I

(11.3.14) Jf[E,gr] and [E',gr'] are topological'groups, 61,fb and'U'h are the corresponding bilateral uniformities, and h: E _,, E' is a (gr, gr')-continuous .homomorphism, then h is (61£b, 'U' h)-uniformly continuous. Proof. By (11.2.17) h is (61£•, 6U 18 )- and ('Ud, 6U 'd)-uniformly continuous (with the usual notations). From this, on account of (3.2.53), it follows that his ('Uh, 61£'8)and (61,fb, 6U 'd)-uniformly continuous and is thus (';Ub, 6U ' h)-uniformly continuous by (3.2.54).1

In particular, an isomorphism which is a (gr, gr,)-homeomorphism is at the same time a (61,fb, 'U' h)-unimorphism. 11.3.d. The completion of a topological group. It will be shown that a separated topological group equipped with an admissible uniformity can b.e extended to a complete topological group.

(11.3.15) Let [E, Sf] be a separated topological group, 6U an admissible uniformity on E, [E', 61£'] the completion of the uniform space [E, 'U]. Then an operation TC 1 : E' xE' _,, E' can be defined on E' in a unique way so that E' equipped with TC 1 i's a group, and [E', Sf'!.!,] is a topological group of which Eis a subgroup. Proof. Let gr, = ·gr'l.t'· If TC 1 has the enumerated properties, then it is (gr' x gr,; $')-continuous and TC 1 I Ex E =TC.As Eis W'-derise, thus ExEis W' x8T'-dense by (7.1.27) and gr, is a T2-topology by (6.3.29), because 6U is separated, therefore these properties define TC 1 by (6.2.3) in a unique manner a:s the only continuous extension of TC to E' xE'. It will be shown now that under our hypotheses TC has indeed a (W' x W', W')continuous extension. In fact, if (x, y) EE' xE' and ro'(x, y) denotes the W' ~ W'neighbourhood filter of the point (x, y), while b'(x) and b 1(y) are the W'-neighbourhood filters of x and y respectively, then by (7.1.22)

ro' = {V{ x V2: V{ E b'(x),

Vi E b'(y)}

is a filter base equivalent to ro'(x, y), therefore W'(x, y)(n){E xE} ~ ro' (n){ExE} holds in view of (2.1.17) and evidently

ro'(n) {ExE} = {(V{nE)x(V~nE): V{E b'(x), V2E b'(y)} =

= {Vix: V2 :

V1 E b'(x) (n) {E},

Vi E b'(y) (n) {E} }.

464

11.3.

COMPLETE GROUPS

(11.3.16)

Now tJ'(x) and tJ'(y) are Sf'-convergent, and thus are 611'-Cauchy filters in view of (5.1.1). Hence tJ 1 = tJ'(x) (n) {E} and tJ 2 = tJ'(y) (n) {E} are 611'-Cauchy filter bases by (5.1.7) and at the same time 611-Cauchy filter bases on account of (5.1.6). Therefore n(tu' (n) {E xE}) = tJ 1 tJ 2 is also a 611-Cauchy filter base and at the same time a 611'-Cauchy filter base, so that n(tu' (n) {ExE}) is ST'-convergent and then together with it n(tv'(x,y) (n){E xE}), equivalent to it by (2.6.7), is Sf'-convergent as well. This guarantees however by (6.2.2) the existence of a (Sf' x ST', ST')-continuous extension n' of n. A similar reasoning shows that the map t: E - E has a (Sf', ST')-continuous extension t' since, with the preceding notation, if x EE' then t(tJ'(x) (n){E}) = = t(tJ1) = tJ1 1 is a 611-Cauchy filter base and hence is Sf'-convergent. In order to see that£', equipped with the operation n', is a group, we show that if x, y, z EE', then n' (n'(x, y), z) = n'(x, n'(y, z)), n'(e, x) = n'(x, e) = x, n'(t'(x), x) = n'(x, t'(x)) = e.

All these follow from (6.2.3) as all three equalities are valid if x, y, z EE (in which case n can be written instead of n ', t instead oft') and both sides of the first equality define($' x3f' x3f',3f')-continuous mapsfromE' xE' xE' into £',further all three members of the second and the third equality define($', $')-continuous maps from E' into itself. For example, we verify that f: E' xE' xE' - E' is ($' x x $' x $', W)-continuous for f(x, y, z) = n' (n'(x, y), z). If Pi, p 2, Pa denote the projections of the product£' xE' xE' on its factors, q: E' xE' xE' -E' xE' is the map for which q(x, y, z) = (x, y), while r: E' xE' xE' -E' xE' is given by the formula r(x, y, z) = (n'(x, y), z), then in view of the ($' x $' x $', ST')-continuity of Pi, p 2, Pa and by (7.1.28), q is ($' x $' x $', $' x $')-continuous, thus n' o q is (ST' x $' x ST', $')-continuous by (2.6.15) · hence r is (ST' x ST' x $', ST' x $')-continuous, finally f = n' o r is ($' x $' x 3i', ST')-continuous. The other asserted continuities can be proved in the same way. [E', ST'] is a topological group because n' is($' x3f', $')-continuous by definition, while t' -::- which gives theinverse in E' - is($', $')-continuous. I (11.3.16) Under the hypotheses of (11.3.15), if Eis a commutative group, then so is E' as well. Proof. With the previous notation, x, y EE' implies n'(x, y) = n'(y, x) again as a consequence of (6.2.3) as both terms define (ST' x ST', ST')-continuous maps and the equality holds for x, y EE. I (11.3.17) Under the hypotheses of (11.3.15), if the left, right, or bilateral uniformity of [E', ST'll,] is 611'', 611'd, or611'b respectively, then the corresponding uniformity of [E, 3i] is 611'• IE, 611'd IE, 611'b IE. Proof. It is a consequence of (11.2.11) and (11.3.13).1

(11.3.20)

11.3.d.

THE COMPLETION OF A TOPOLOGICAL GROUP

465

We understand by a completion of the separated topological group [E, 8T] a separated topological group [E', 8f'J which is complete and whose [E, 8T] is a dense subgroup. For every separated topological group there can b~ found a completion. (11.3.18) Let [E, 8!] be a separated topological group, "l,l'.b its bilateral uniformity, and [E', 8!'] the topological group constructed by (11.3.15) with the choice "U'. = "l,l'.b. Then [E', 8!'] is a completion of [E, 8!]. Proof. If "Ll'. = "l,l'.b then (11.3 .15) can be applied as indeed "l,l'.b is admissible by (11.3.9). Then 61,C' is a separated complete uniformity such that 8J' = 8Jq,e,, Eis 8T'-dense and "Ll'.' IE= "U'.b. On the other hand, by (11.3.17), for the bilateral uniformity "U'.' h of the group [E', 8!'] we have 8J' = 8J'Lf'b and "Ll'.' h / E = "l,l'.b. Therefore by (6.3.1) 6U' = "U'.' h so that the bilateral uniformity of [E', 8!'] is complete. I (11.3.19) lf [E', 8T'] and [E", 8T"] are completions of the separated topological group [E, 8!] then there exists a unique isomorphism h: E' ~ E" fixing E which is a (8!', 8!")-homeomorphism. Proof. Let "Ll'.' and "Ll'." be the bilateral uniformities of the groups [E', 8!'] and [E", 8T"] respectively. By (I 1.3.17) both [E', "U'.'] and [E", "Ll'."] are completions of the uniform space [E, "l,l'.b] where "l,l'.b denotes the bilateral uniformity of the group [E, 8!], and hence there exists a unique (6U', "Ll'.")-unimorphism h fixing E by (6.3.26). This map his at the same time a (8T', 8J")-homeomorphism and - as it is bijective - an isomorphism. Indeed if x, y EE' n"(h(x), h(y))

= h(n'(x, y)),

for the two maps which map the point (x, y) EE' xE' to the left and the right hand sides of the equality are (8T' x 8!', 8J")-continuous and the equality holds for (x, y) EE xE (it goes over to n(x, y) = n(x, y)), so that (6.2.3) can be applied. Conversely every isomorphism h1 : E' ~ E" fixing E which is a (8J', 8J")-homeomorphism is by (11.3.14) a ("Ll'.', "Ll'.")-unimorphism, and thus is identical with the above map h. I (11.3.20) The completion [E', 8!'] of the separated topological group [E, 8l] is left complete if! the left uniformity of [E, 8T] is admissible. Proof. If the left uniformity "U'.'s of [E', 8J']is complete, then the left uniformity "Us = "U'. 's / E of [E, 8!] is admissible by (11.3.17) and (11.3.1 ). Conversely suppose that "Us is admissible and apply (11.3.15) by choosing "Ll'. = ="Us.Then we obtain a topological group [E", 8)'11.] and the complete uniformity "Ll'." is an extension of"U'.s. Since by (11.3.17) the left uniformity "U'."s of [E", W'li"] is also an extension of "Us and 8J'Lf"s = 8)'11", therefore by (6.3.1) "U'. 118 = "U'." and [E", 8T'li"] is left complete. Hence [E", 8!'11"] is complete by (11.3.10) and as [E, 8!] is a dense subgroup of [E", 8)'11,, ], therefore [E", 8T'li.] is a completion of [E, 8!]. By (11.3.19) [E',8!'] and [E",8Jq,e.] are isomorphic which implies at once that [E', 8T'] is left complete as well. I We can add to (11.3.20) by (11.3.1) and (11.2.11) that if there exists at all a left 30 Akos Csaszar

466

11.3.

COMPLETE GROUPS

(11.3.21)

complete topological group [E', §'] whose [E, §J is a (not necessarily dense) subgroup, then the left uniformity of [E, §] is admissible. 11.3.e. Locally compact groups. We have seen that the topological group investigated at the end of 11.2.b is left complete; this follows already from the fact that its topology is locally compact (as we have to do with an open subspace of the space [R2, ~ 2 ]): (11.3.21) Every locally compact topological group is left complete. Proof. Let V be a compact neighbourhood of e in the topological group [E, §], t a 6113 -Cauchy filter base, R 0 E t small of order Uf,, x E R 0 • Then y E R 0 implies x- 1y E V, thus y E x V, R 0 c x V, and by (11.2.1) x Vis compact as well. Hence the filter base t 0 = t (n) {R0 } has a cluster point in x V; this is also a cluster point of t by (5.2.26) and then tis convergent by (5.2.28). I

The question arises which topological groups have locally compact or even compact completions. In connection with this let us note first that: (11.3.22) In any topological group [E, §J the following statements are equivalent:

(a) 6)1 3 is precompact; (b) 6Ud is precompact; (c) 6)1h is precompact. Proof. (a)= (b): We refer to (5.2.18). If t is ®'.,,rd-compressed, then t being (&6Ud, &61,fs)-proximally continuous by (11.3.3), x- 1 = t(t) will be Sl'"l,fs-compressed on account of (5.2.2). Thus it will be a 6118 -Cauchy filter base and then by (11.3.5) t will be a 6Ud-Cauchy filter base. (b) = (a): In the same way. (a) and (b) = (c): (3.2.75). (c) = (a) and (b): (3.2.74).1

The topological group [E, §] or the set A c Eis said to be precompact if it is precompact with respect to 6)1h. Then by (6.3.31), (11.3.18), and (11.3.19) we can say that: (11.3.23) The completion of a separated topological group is compact is precompact. I

if! the group

(11.3.24) The completion of a separated topological group is locally compact if! e has a precompact neighbourhood in the group. Proof. If [E', §'] is the completion of the topological group [E, §J and V' is a compact §'-neighbourhood of e, then V' n E is a 8f-neighbourhood of e and is 6)1'b-precompact by (5.2.21) and (3.2.70), i.e. it is 61,Ch-precompact by (11.3.17).

Conversely if Vis 61,Cb-precompact,i.e. a 6U'b-precompact 8!-neighbourhood of e, then its 8f'-closure Vis 6U'h-precompact as well by (3.2.76); it is also 6U'b I V-complete by (5.1.15) so that Vis §'-compact on account of (5.2.20). If G is a 8f'-open set such that e EG n E c V, then in every §' -neighbourhood of every point x E G there is a point of E - as Eis §' -dense - thus a point of V, i.e. x E V, G c V,

11.3.e.

(11.3.26)

467

LOCALLY COMPACT GROUPS

and Vis a 8f'-neighbourhood of e. Hence e has a compact gr,-neighbourhood, and by (11.2.1) this holds for any other point of E' as well. I (11.3.25) If in a topological group A and Bare compact, then AB is again compact. Proof. AB= n(A xB) so that the statement follows from (7.1.42) and (5.3.10), I A further important property of locally compact groups is the following: (11.3.26) Every locally compact topological group is paracompact. Proof. Let in the topological group [E, 8T] Vbe a compact closed neighbourhood of e (this exists by (5.3.54)). If V0 = V n v- 1, and n EN implies Vn ,= Vn-l Vo, then in view of (11.2.1) v- 1 is compact, closed, V0 as its closed subset is compact by (5.3.4), and from this the compactness of Vn follows for every n on account of 00

(11.3.25). Now E 0

= U Vn

is a subgroup as x E Vm y E Vm implies xy E Vn+m

0

oo

and if X E Vn then x- 1 E Vn. Eo is open-closed by (11.2.12) so that Eo = -

.

u vn holds 0

as well and since Vn is also compact for every n on account of (5.3.20), therefore E0 is a-paracompact by (8.3.20),and since it is regular, it is paracompact by (8.3.15). As a consequence of (11.2.1) the same holds for the cosets xE0 and then the statement follows from (8.3.23) by (11.1.6), I 11.3.f. Exercises. 1. Show that, for the topological group [R, &$] - where n(x, y) = x + y - the finest uniformity 611 inducing &$, distinct from the invariant uniformity 611b = 611P,' is admissible. [611b is complete, thus since 611b < 611 so is 611.] 2. Let the operation n(x, y) = x + y be defined on Q and gr = &$ I Q. Show that (a) The topological group [Q, gr] is not complete; (b) The group [R, &$] in 1. is the completion of [Q, gr]; (c) gr can be induced by a complete uniformity 611.

[(6.4.36).] 3. Let [E, gr] be a topological group, 611 a uniformity inducing 8J, and suppose that n is (611 x611, 611)-uniformly continuous. Show that (a) To every surrounding U E 611, we can find a surrounding U1 E 611 such that (x1, yJ E U1, (x2, yJ E U1 iniply (x1X2, YiYJ E U; (b) If Uand U1 fulfil (a) and V = U(e), then U1 c Uv n Ut; (c) If U and U1 satisfy (a), then, with the notation V1 = U1(e), we have Uv, u u Ut, CU; (d) 6118 = 6114 = 611, 4. Let Ebe a group, p a left invariant pseudo-metric on E. Show that (a) and (b), and similarly (c) and (d), are equivalent to each other: (a) grP is a compatible topology of E; (b) For a EE and 8 > 0 there is a 8 > 0 such thatp(a, x) < 8 implies p(a- 1, x- 1) < B; (c) 8fPis a compatible topology of E and in the topological group [E, 8fp] we have 6118 = 6114 ; 30*

11.3,

468

COMPLETE GROUPS

(d) For s > 0, there is a c5 > 0 such that p(x, y) < c5 implies p(x- 1, y- 1) < e. [(11.2.3) (a)-(d) are fulfilled for the system of the balls S(e, e) while (e) is equivalent to (b) here. If SfPis compatible then 611P = 6118 . ] 5. Let Ebe the set of all bijections of the interval I onto itself, n(f, g) = f o g, and p(f, g) = sup { lf(t) - g(t) I : t E I}. Show that (a) p is a right invariant metric on E; (b) If

fit)-[:-

= 0), (0 < t < I), (t

t

(t = 1),

further 1 n

(t

1- t

(o < t < I - !J,

fnCt) =

t -

= 0),

1+- (1-! ~tB AuB

U{A;: i EI} LJ A;

1.1.a. 1.1.a. 1.1.a. 1.1.a. 1.1.a. 1.1.b. 1.1.b. 1.1.c. 1.1.c. 1.1.c.

x belongs to the set A x does not belong to the set A

empty set the set consisting of the elements a, b, c the set of the elements x having the property P(x) A is a subset of B A contains B as a subset union of A and B

union of a family of sets union of a family of sets

iEJ n

LJ

A,

1.1.c.

union of a finite number of sets

LJ A,

lJ.C;

union of a sequence of sets

AnB n {A 1: i EI} A,

1.1.c. 1.1.c. 1.1.c.

intersection of A and B intersection of a family of sets intersection of a family of sets

1.1.c.

intersection of a finite number of sets

1.1.c.

intersection of a sequence of sets

1.1.c. 1.1.e. 1.1.e. 1.1.e. 1.1.e. 1.1.e. 1.1.e. 1.1.e. 1.1.e. 1.1.e. 1.1.e. 1.1.e.

difference of A and B the set of the real numbers the set of the natural numbers the set of the rational numbers open interval closed interval interval closed on the left interval closed on the right open interval infinite on the right interval infinite on the right, closed on the left open interval infinite on the left interval infinite on the left, closed on the right

m 00

m

n nm A; nm A,

iEJ n

00

A-B R

N Q (a, b) [a, b] [a, b) (a, b] (a, +m) [a,+ m) ( - CXJ, a) (-m,a]

NOTATIONS

1.1.e. 1.1.e. 1.1.e. 1.1.e. 1.1.e. [xi 1.1.e. max (x1, ••• , Xn) 1.1.e. min (x 1, ••• , Xn) 1.1.f. (an) 1.1.f. lim an 1.1.f. an --+ a 1.1.g. (a1, .. •, an) Rm 1.2.a. p(x, y) .1.2.a. 1.2.b. S(a, e) 1.2.b. Xn--+ y 1.2.b. Iimxn c5(A) 1.2.b. (a1, ... , am; b1, ... , bm) 1.2.c. [a1, . .. , am; b1, . .. , bm] 1.2.c. S(a, e) 1.2.c. (Di), ... , (DJ 1.3.a. [E, p] 1.3.a. 1.3.a. Pm 1.3.a. P JEo p(x, A) 1.3.e. p(A, B) 1.3.e. d(A,B) 1.3.e. 2.1.b. m< ?S 2.1.b. m> ?S m,.__, ?S 2.1.b. A 2.1.b. x 2.1.b. m(n) ?S 2.1.b. m(u) ?S 2.1.b. [E, ~] 2.1.c. tJ(x) 2.1.c. 2.1.d. Xn--+ y 2.1.d. lim Xn grp 2.1.e. © 2.1.e. ©m 2.1.e. @+ 2.1.e. ©2.1.e. (V), (V'), (V") 2.2.a.

(-CXl,+CXl)

I sup A inf A

31*

483

the real line the interval (0, 1] least upper bound of a set of real numbers greatest lower bound of a set of real numbers absolute value of a real number the greatest of x 1 , ••• , Xn the least of x 1 , • •• , Xn sequence limit of a numerical sequence limit of a numerical sequence m-sequence m-dimensional Euclidean space distance ball limit of a sequence of points limit of a sequence of points diameter of a point set open brick closed brick closed ball metric axioms (pseudo-)metric space Euclidean metric in Rm restriction of a (pseudo-)metric distance of a point and a set distance of two sets Hausdorff distance mis coarser than ?S mis finer than ?S mand ?S are equivalent principal filter associated with A fundamental filter associated with x system of intersections system of unions neighbourhood space neighbourhood filter limit of a sequence of points limit of a sequence of points topology induced by a pseudo-metric Euclidean topology of the real line Euclidean topology of Rm topology of the convergence on the right topology of the convergence on the left neighbourhood axioms

484

NOTATIONS

(~, x)

2.2.c. 2.2.c. 2.2.c. 2.2.c. 2.2.c. 2.2.d. 2.2.d. 2.2.d. 2.2.e. 2.2.f. 2.2.g.8.

(~, x]

2.2.g.8.

(x, --+)

2.2.g.8.

[x, --+)

2.2.g.8.

(a, b) [a, b) (a, b]

2.2.g.8. 2.2.g.8. 2.2.g.8. 2.3.a. 2.3.a. 2.3.a. 2.3.a. 2.3.a. 2.3.b. 2.4.a. 2.4.b. 2.4.b. 2.4.c. 2.5.a. 2.5.b. 2.5.c. 2.5.c. 2.5.c. 2.5.d. 2.5.d. 2.5.e. 2.5.e. 2.5.f. 2.5.f. 2.5.g. 2.5.g.

(G1), (G2), (Ga) CF1), (F2), (Fa) & & g}E

int A

A (K1), ... , (KJ . mar A (H1), ... , (HJ

~l ~l

< ~2 > ~2

®e inf {8l";: i EI} sup {8f;: i E/} ~IEo 8)5'

t--+ x

X (~)

= lim'll't

(M1), (M 2) 1:J(A) (To) (S1)

x (TJ (S2) (T2) (Sa) (Ta) (SJ (TJ (85) (Ts)

axioms of the open sets axioms of the closed sets topology of the semi-continuity from above topology of the semi-continuity from below topology of the finite-closed sets interior of A closure of A Kuratowski closure axioms boundary of A Hausdorff neighbourhood axioms open interval infinite on the left in an ordered set interval infinite on the left, closed on the right in an ordered set open interval infinite on the right in an ordered set interval infinite on the right, closed on the left in an ordered set open interval in an ordered set interval closed on the left in an ordered set interval closed on the right in an ordered set ~1 is coarser than ~2 ~1 is finer than ~2 discrete topology infimum of topologies supremum of topologies restriction of a neighbourhood structure topology refined by an ideal limit of a filter base in a neighbourhood space limit of a filter base in a neighbourhood space axioms of countability neighbourhood filter of a set separation axiom separation axiom closure of -a set consisting of a single point separation axiom separation axiom separation axiom separation axiom separation axiom separation axiom separation axiom separation axiom separation axiom

NOTATIONS

f: X---+ Y f(x) f(A)

U(a)

2.6.a. 2.6.a. 2.6.a. 2.6.a. 2.6.a. 2.6.a. 2.6.a. 2.6.a. 2.6.a. 2.6.b. 2.6.b. 2.6.f. 3.1.b. 3.1.b. 3.1.b. 3.1.b. 3.1.b. 3.1.b. 3.1.c. 3.1.d. 3.1.d. 3.1.d. 3.1.d. 3.1.e. 3.1.f. 3.2.a. 3.2.a:. 3.2.b. 3.2.b. 3.2.b.

U(x)

3.2.b.

u-1

3.2.b. 3.2.b. 3.2.c. 3.2.c. 3.2.c. 3.2.d.

1-1

J- 1(A) 1-1(y)

fl 1 !IA gof f(W.)

1-1(~)

1-vn A&B

A&B (P1), · · ., (Pa)

SYP [E, &] ,p(A) gr& &1 < &2 &1 > &2 sup {&;: i EI} inf {&;: i EI} & I Eo

1-1(&) UP,•

u.

AxB Gf

VoU (UJ, ... , (UJ

6Up [E, 6\1] Ui:,e

3.2.d. 3.2.e.

485

map from X into Y image of an element image of a set inverse map inverse image of a set inverse image of an element restriction of a map restriction of a map composition of maps image of a system of sets inverse image of a system of sets inverse image of a topology A is near to B A is far from B proximity axioms proximity induced by a pseudo-metric proximity space proximity filter of a set topology induced by a proximity &1 is coarser than &2 &1 is finer than &2 supremum of proximities infimum of proximities restriction of a proximity inverse image of a proximity e-surrounding in a pseudo-metric space e-surrounding in a pseudo-metric space Cartesian product of two sets graph of a map set consisting of the second members of the pairs belonging to U with first members from A set consisting of the second members of the pairs belonging to U with first member x set of tte pairs from U with inverse order composition of two sets consisting of pairs uniformity axioms unifon;nity induced by a pseudo-metric uniform space e-surrounding associated with a family of pseudo-metrics uniformity induced by a family of pseudometrics proximity induced by a uniformity

486

NOTATIONS

8f""ll &x

3.2.e. 3.2.e.

8f"X 611'.1 < 611'.2 611'.1 > 611'.2 sup {611'.1: i E J} inf {611'.1: i E I} 6U'. \ Eo