Royal Road to Topology, A: Convergence of Filters 9811232105, 9789811232107

Table of contents :
Table of Contents
About the Author
Preface
Chapter I. Preliminaries
I.1. Relations and maps
I.2. Equivalence relations
I.3. Order
I.4. Polarities
I.5. Finite and infinite cardinals
Chapter II. From convergence of sequences to the concept of filter
II.1. Convergence of sequences
II.2. The concept of filter
II.3. Order on filters
II.4. Decomposition into free and principal filters
II.5. Supplement
Chapter III. Convergence of filters
III.1. The concept of limit
III.2. A pointwise perspective
III.3. Order on convergences
III.4. Prime convergences
III.5. Finite convergences
III.6. Metrizable convergences
III.7. Supplement
Chapter IV. Continuity
IV.1. Images and preimages
IV.2. Continuous maps
IV.3. Continuity and order
IV.4. Initial convergences
IV.5. Selections and products
IV.6. Product convergences
IV.7. Diagonal product maps
IV.8. Final convergences
IV.9. *Lower semicontinuity
IV.10. Supplement
Chapter V. Families of sets
V.1. Isotone and antitone families
V.2. Ideals, filters, distributive families
V.3. The notion of grill
V.4. Ultrafilters
V.5. Various classes of filters
V.6. Supplement
Chapter VI. Pretopologies
VI.1. Basic facts about pretopologies
VI.2. Adherence
VI.3. Inherence
VI.4. Continuity
VI.5. Structural aspects of pretopologies
VI.6. Pretopologizer
VI.7. Initial density of the Bourdaud pretopology
VI.8. Supplement
Chapter VII. Topological structures
VII.1. Closed sets, closure
VII.2. Open sets, interior, neighborhoods
VII.3. Topologies
VII.4. Structure of the class of topologies
VII.5. Initial density of the Sierpinski topology
VII.6. *Remarks on convergence of nets
VII.7. Supplement
Chapter VIII. Adherences, covers, and compactness
VIII.1. Adherences
VIII.2. Covers and inherences
VIII.3. Compact convergences
VIII.4. Supplement
Chapter IX. Topological concepts
IX.1. Topological separation
IX.2. Bases
IX.3. Compact, countably compact, and Lindelöf topologies
IX.4. Open, closed, topologically quotient, and perfect maps
IX.5. Supplement
Chapter X. Functional study of topologies
X.1. The real line
X.2. Real-valued functions
X.3. More on metrics and metrizable topologies
X.4. Completely metrizable topologies
X.5. Completion
X.6. Functionally closed and open sets
X.7. Functional separation
X.8. *A non-trivial regular topology with each continuous function constant
X.9. Functionally initial topologies
X.10. Supplement
Chapter XI. Functional partitions and metrization
XI.1. Infinite sums of positive real numbers
XI.2. Partitions of unity
XI.3. Paracompactness
XI.4. Metrization theorems
XI.5. Supplement
Chapter XII. Compact topologies
XII.1. A short story of compactness
XII.2. Compact sets in topological spaces
XII.3. Weaker variants of compactness
XII.4. Sequentially compact topologies
XII.5. The Cantor set
XII.6. *Cardinality aspects
XII.7. The Stone topology
XII.8. *Almost disjoint families
XII.9. *Hyperconvergences and semicontinuities
XII.10. Supplement
Chapter XIII. Connected and disconnected topologies
XIII.1. Connected topologies
XIII.2. Components and quasi-components
XIII.3. Locally and arcwise connected topologies
XIII.4. Disconnected topologies
XIII.5. Ultrametric and ultrametrizable spaces
XIII.6. Cantor-Bendixon theorem
XIII.7. Supplement
Chapter XIV. Extensions and compactifications
XIV.1. Extensions of spaces and maps
XIV.2. Compactifications
XIV.3. Filters based in lattices
XIV.4. Cech-Stone compactification
XIV.5. A construction of maximal compactifications
XIV.6. Supplement
Chapter XV. Uniform structures
XV.1. Preuniformities
XV.2. Uniformities
XV.3. Operations
XV.4. Completeness
XV.5. Supplement
Chapter XVI. Sequentially founded convergences
XVI.1. Sequential filters
XVI.2. Subsequences and subquences
XVI.3. Sequentially founded convergences
XVI.4. *Extrema of sets of sequential filters
XVI.5. Supplement
Chapter XVII. Structural aspects
XVII.1. Projective and coprojective classes
XVII.2. Functors
XVII.3. Reflective and coreflective classes
XVII.4. Initial and final functors
XVII.5. Supplement
Chapter XVIII. Fundamental classes
XVIII.1. Stability and pavements
XVIII.2. Foundations
XVIII.3. Pseudotopologies
XVIII.4. *Initial density, exponential hulls
XVIII.5. Adherence-determined convergences
XVIII.6. Paratopologies
XVIII.7. Hypotopologies
XVIII.8. Commutation of functors with products
XVIII.9. Classification of quotient maps
XVIII.10. Supplement
Chapter XIX. Diagonality and regularity
XIX.1. Contour operation
XIX.2. Diagonality
XIX.3. Regularity
XIX.4. *Partial regularization
XIX.5. *P-points
XIX.6. Supplement
Chapter XX. Compactness
XX.1. Compactness versus cover-compactness
XX.2. From pseudotopologies to compactness
XX.3. Conditional compactness
XX.4. Countable and sequential compactness
XX.5. Lindelöf property
XX.6. Compactoid and compact families
XX.7. Conditional compactness of products
XX.8. Regularity and topologicity in compact pseudotopologies
XX.9. Local compactness
XX.10. Hyperconvergences revisited
XX.11. Classification of perfect maps
XX.12. Compactness related properties of maps
XX.13. Supplement
Chapter XXI. Mixed properties
XXI.1. Fréchet pretopologies
XXI.2. Galois connections
XXI.3. Functorial equations and inequalities
XXI.4. Sequentially induced convergences
XXI.5. Sequentially founded convergences induced by adherence-determined classes
XXI.6. Subclasses induced by locally compact convergences
XXI.7. Preservation of mixed properties under quotient maps
XXI.8. Preservation of mixed properties under products
XXI.9. *Relations and polarities in classes of filters
XXI.10. Coreflectors for mixed properties
XXI.11. Supplement
Chapter XXII. Implementations and refinements
XXII.1. Active boundary
XXII.2. *Topological defect and sequential order
XXII.3. *Sequential cascades
XXII.4. Subsequential topologies
XXII.5. Topologically maximal pretopologies
XXII.6. Applications of the Stone topology
XXII.7. *Homogeneity quest
XXII.8. *A compact Fréchet topology with non-Fréchet square
XXII.9. Measure-theoretic convergences
Chapter XXIII. Completeness
XXIII.1. Fundamental filters and completeness
XXIII.2. Cocompleteness
XXIII.3. Finitely complete convergences
XXIII.4. Countably complete convergences
XXIII.5. Completeness number
XXIII.6. Completeness of subspaces
XXIII.7. Operations preserving completeness number
XXIII.8. Baire category theorem
XXIII.9. *Ultracompleteness
XXIII.10. Supplement
Chapter XXIV. Spaces of maps
XXIV.1. Dual (natural) convergence
XXIV.2. Exponentiation, transposition, and adjoint maps
XXIV.3. Exponential law
XXIV.4. Hyperconvergences as dual convergences
XXIV.5. Other convergences on function spaces
XXIV.6. A unification of functional convergences and hyperconvergences
XXIV.7. Consonance
XXIV.8. Compactness in function spaces
Chapter XXV. Duality
XXV.1. Dual and bidual convergences
XXV.2. Bidual reflector
XXV.3. Reflective inequalities in duality theory
XXV.4. Exponential classes and hulls
XXV.5. Functionally embedded convergences
XXV.6. Epitopologies
XXV.7. Duality in completeness
Chapter XXVI. Modified duality
XXVI.1. Functorial modifications of duality
XXVI.2. Applications to mixed properties
XXVI.3. Exponential objects
XXVI.4. Topologicity of dual convergences
XXVI.5. Conditional versions of core compactness
Chapter XXVII. Outline of principles
XXVII.1. Axioms of set theory
XXVII.2. Cardinality
XXVII.3. Well-ordered sets
XXVII.4. Ordinal numbers
XXVII.5. Ordinal arithmetic
XXVII.6. Cardinal ordinals (Alephs)
XXVII.7. Closed unbounded sets
XXVII.8. Well-founded relations
XXVII.9. Small cardinals
XXVII.10. Supplement
Chapter XXVIII. Category theoretic perspective
XXVIII.1. Objects, morphisms, and functors
XXVIII.2. Constructs
XXVIII.3. Topological categories
XXVIII.4. Reflections and coreflections in concrete categories
XXVIII.5. Conclusion
Postface
Bibliography
List of Symbols
Index

Citation preview

A Royal Road to Topology Convergence of Filters

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A Royal Road to Topology Convergence of Filters

Szymon Dolecki Mathematical Institute of Burgundy, Dijon, France

World Scientific NEW JERSEY

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LONDON

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TOKYO

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Names: Dolecki, Szymon, author. Title: A royal road to topology : convergence of filters / Szymon Dolecki, Mathematical Institute of Burgundy, France. Description: New Jersey : World Scientific, [2024] | Includes bibliographical references and index. Identifiers: LCCN 2022013688 | ISBN 9789811232107 (hardcover) | ISBN 9789811232114 (ebook for institutions) | ISBN 9789811232121 (ebook for individuals) Subjects: LCSH: Topology. | Convergence. Classification: LCC QA611 .D655 2024 | DDC 514--dc23/eng20220621 LC record available at https://lccn.loc.gov/2022013688 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Cover: A. F. Van der Meulen, Ingress of Louis XIV and Marie-Thérèse to Arras on 30 July 1667, Musée des Beaux-Arts, Arras. Photo (C) RMN-Grand Palais / Christian Jean Copyright © 2024 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

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Desk Editors: Nandhakumar Krishnan/Tan Rok Ting Typeset by Stallion Press Email: [email protected] Printed in Singapore

About the Author Szymon Dolecki was an Emeritus Professor at the Mathematical Institute of Burgundy, Dijon, till 2023. He received his PhD in 1973 from the Institute of Mathematics, Polish Academy of Sciences, where he was employed till 1984. He was a Professor at the University of Limoges (1985–1990) and Burgundy University (1990–2017). He is the author of some hundred publications in convergence theory, topology, optimization, control theory, functional analysis, and history of mathematics. His books are as follows: Analyse Fondamentale, Hermann, Paris, 2013 (2nd edition), Convergence Foundations of Topology, Frédéric Mynard (coauthor), World Scientific, 2016.

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Preface Topological spaces are a special case of convergence spaces. This textbook introduces topology within a broader context of convergence theory. It comprises the matter usually taught in courses of general topology (Chapters VII to XV). The book is addressed both to those who wish to learn topology from the start, and to those who, being already knowledgeable, are curious to review topology from a different perspective, which goes well beyond the traditional knowledge. In particular, a topologist will find some arguments (1) that become much simpler due to the approach adopted in this book. As for the beginners, to start up reading this book does not require any knowledge about topologies, not even about metric spaces. It would be enough to have studied convergence of sequences on the real line (2), and if you are not familiar with convergence of sequences, which may be the case of those who have studied calculus without sequences, start to read the second chapter by Section II.2. Nevertheless, some level of general mathematical knowledge is presumed, namely familiarity with order, set-theoretic operations, maps, quantifiers, and so on. We use sequences as a motivation for filters. Having lost their foregoing importance, sequences remain a useful elementary tool to substantiate certain abstract facts involving filters.

1Let

me mention Stone’s theorem on paracompactness of metrizable topologies (with a non-technical proof of few lines). 2I realize, however, that this seemingly innocuous sentence can have different meaning for a European and an American reader. A traditional European way of teaching analysis is to start by a study of sequences and their convergence on the real line. A typical calculus course in the U. S. A. attains differentiation and integration without mentioning sequences, which is quite an achievement. Indeed, it is hard to conceive for those from the other tradition that it is possible to do this without sequences. The point is that it is feasible, but without defining rigorously the notions of limit and convergence. In other terms, one can be impressively good in calculating derivatives and integrals, without need of knowing what they really are. vii

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PREFACE

Convergence of filters is a natural generalization of convergence of sequences, so that convergence theory is more natural and intuitive to many, perhaps most, students, rather than topology courses based on open and closed sets (3). The framework of convergence is more general than that of topology, and it is exactly the proper level of generality. By a “proper level of abstraction” with respect to (an objective of understanding of) a given theory, I mean the level, which is operative on one hand, and intuitive on the other. Of course, this level depends on a student’s background, that comprises experience and cognitive pattern, but one can bet that there is some dose of universality in the concept. If an approach is very abstract, its sense is often obscure, and its application arduous; if it is not sufficiently general, its arguments are usually laborious and, most importantly, lacking enough insight. A knowledgeable reader will recognize some footprints of the category theory, outlined in Chapter XXVIII at the end of the book. I refrain from its formal introduction in the body of the book in order not to increase the level of abstraction more than necessary, that is, an objectwise application of concrete categories. This objectwise approach 3Filters

are either marginal or inexistent in European classics, like [235] of W. Sierpiński, [169] of K. Kuratowski, and [97] of R. Engelking, as well as in American standards, like [147] by J. Kelley, [94] by J. Dugundji, and [191] by J. R. Munkres. The collective oeuvre signed N. Bourbaki is a vast compendium of mathematics. In the volumes dedicated to General Topology [31] filters are used, but standing back, secondary with respect to open and closed sets. Nevertheless, occasionally they are playing the first roles, like in the presentation of compactness. The book of L. Gillman and M. Jerison [113] constitutes an exception to the low profile of filters, though it is both advanced and specialized to the rings of continuous functions and compactifications of completely regular topologies. None of these books goes beyond the topological framework. There exists, however, a classical book [40] of E. Čech, which is developed in a greater generality of pretopologies, in terms of non-idempotent closure. W. Gähler’s [107] makes a real difference, but is largely unknown — I believe that it is because of its high level of abstraction and perhaps also because it is written in German, which is not widely known nowadays. There are similarities between the book of Gähler and the present one, like use of filters and other relevant families of sets, and structural foundation based on categories and functors. In both, topologies materialize after some 150 pages have been devoted to other themes. We have a common motivation, that is, to base the understanding of topology and analysis on the versatility of the convergence theory, which compensates shortcomings of topology. A difference lies in an abstract, formal style of the former. It is significant that settheoretic and category foundations constitute the opening of W. Gähler’s book, and the appendices of mine.

PREFACE

ix

to categories is an instance of a more general way of looking at things, that is, adapting the tools to the setting in which they are used. Filters prevail over nets in the context of convergence, because, from the convergence viewpoint, each net can be reduced to the filter it determines, and, moreover, because the calculus of nets is intricate in contrast to that, easy, of filters (see Section VII.6). I have already argued that convergence theory is intuitive. On the other hand, it is very operational in the topological context, because the class of convergences is closed under several natural and essential operations, under which the class of topologies is not! The class of convergences completes the class of topologies in an analogous way as the field of complex numbers algebraically completes the field of real numbers. I have been repeating the sentence above, citing myself again and again, but I believe that it grasps the essence. Indeed, various topological quests have solutions only beyond the class of topologies (4). Many others can be tackled more straightforwardly and gain on insight when reformulated in convergence terms (5). This sort of fulfillment makes convergence theory far-reaching and easier to comprehend and to implement than topology. Often, it enables a more conscious approach, also to traditional topics of topology. This way we not only observe certain phenomena but also understand why they occur (6). This book attempts to take advantage of these features of convergence theory. It provides essential ingredients of traditional topology courses. But more importantly, it tries to explain the underlying structure and mechanisms. 4For

instance, the coarsest convergence on the set of continuous functions making the natural coupling continuous is always pseudotopological, but not topological, when the underlying topology is not locally compact. 5For instance, it becomes clear that the nature of compactness is not topological, but pseudotopological, when we realize that the Tikhonov theorem is a very special case of the commutation of pseudotopological reflectors with products. Another example is that of mixed properties: the image of a Fréchet topology under a quotient map need not be Fréchet, but it is Fréchet under hereditary quotient maps. This becomes clear from the way in which Fréchet topologies and hereditary quotient maps are defined with the aid of the pretopological modifier. 6For example, it was long well known that a product of two Fréchet topologies need not to be Fréchet. This fact inspired a search for extra conditions under which a given product is Fréchet. However, a structural explication in convergence-theoretic terms of non-productivity of Fréchet property enabled F. Jordan and F. Mynard to characterize those topologies which product with each Fréchet (respectively, strongly Fréchet) topology is Fréchet [141].

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The title of this book alludes to the mentioned advantages of the present approach, which I believe to be more gratifying than many traditional ones: you travel more comfortably and see more. Sire, there is no royal road to geometry! is attributed by some to Euclid in answering Ptolemy I, and by others to Menaechmus in responding to Alexander the Great, when the monarch asked for a quick introduction to geometry. Sure enough, this way to topology is not entirely effortless — otherwise it would be boring — but it offers an attractive trade-off: you invest to grasp the world of filters, but soon you will be gladdened, because this world grants you insight. Perhaps the most significant feature of the class of convergences is exponentiality (7), that is, there always exists a coarsest convergence on the set of continuous maps (from one convergence to another) that makes the natural coupling jointly continuous. This is not the case of topologies! The fact was discovered seven decades ago by G. Choquet (8), founder of convergence theory (9). The phenomenon of non-topologicity of spaces of continuous maps occurs also for dual spaces in the Mackey-Arens duality in functional analysis. Also in this case, methods of convergence theory prove to be very fruitful, as witness Convergence Structures and Applications to Functional Analysis of R. Beattie and H. P. Butzmann and Continuity theory by D. L. Nel (10). We shall tackle metrizability, which has been one of the oldest topics in topology. Thanks to a novel use of partitions of unity proposed by J. Dydak (11), these — traditionally difficult and technical — investigations become accessible and transparent. Alluding to the title of this book, we can say that they constitute a princely road to metrization. The subject of this book is similar to that of Convergence Foundations of Topology by S. Dolecki and F. Mynard (12). I tried, 7Traditionally

called Cartesian closedness. Choquet [44]. 9It is interesting that Choquet qualifies convergence theory as an important part of topology. 10R. Beattie and H. P. Butzmann [24], D. L. Nel [204]. 11J. Dydak [21] and K. Austin, J. Dydak [21]. See also S. Dolecki [64] and S. Dolecki, F. Mynard [83]. 12For practical reasons, I am not always adopting the notation from the previous book [83] by S. Dolecki and F. Mynard. If I realize that the notation used in that book is perhaps heavy or not easy to follow, I try another one. In mathematics, the same symbol may have different meanings in different contexts. One should be aware 8G.

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however, to make it more accessible, by requiring less prerequisites. Its scope is also somewhat altered: while Convergence Foundations of Topology tended to derive most of traditional results from more general facts about convergences, A Royal Road to Topology develops sooner usual classical topological topics, revisiting them later from a conceptually more advanced perspective of convergence theory. These objectives implied a method of recurrent introduction of concepts, starting from examples and special cases and ending with full generality and applications. For accessibility’s sake, this book develops on two levels, fundamental, and more advanced. The sections evidenced by asterisk (*) either require more set-theoretic prerequisites, which are supplied in Chapter XXVII, or use more convoluted arguments, or are otherwise complementary. Most chapters include a section called Supplement, which — as the name indicates — contains some additional facts, as well as some exercises and their solutions. For instance, the fundamental level refrains from discussions of cardinalities others than finite, countable, and uncountable without distinction; the cardinality of continuum and other specific uncountable cardinalities may occur, mostly in asterisked sections. Acknowledgements. I am grateful to Gabriele H. Greco (Trento, Italy), with whom I shared enthusiasm while discovering convergence structures four decades ago. His irreverence to established uses influenced my perception of mathematics and enabled us to transgress the confines of conventional topology. Frédéric Mynard (New Jersey City University) shares a considerable part in shaping convergence theory. I am grateful to him for several important contributions to this book. I am beholden to Alois Lechicki (Fürth, Germany) for unfailing and demanding scrutiny of large portions of the manuscript, which incited me to revise certain of its traits. I am fortunate that Jerzy Wojciechowski (West Virginia University) recently became intrigued by convergence theory. I owe him not only many fine hints and penetrating quests, but also solutions of several open problems. I appreciate the expertise in set theory of Andrzej Starosolski (Silesian Polytechnic, Gliwice, Poland). I am indebted to Ahmed Bouziad (University of Rouen) for having detected a mistake in a preliminary version. that an attempt to build a totally precise and consistent system of notation would lead to very complex symbols. Therefore, such a system would be impracticable.

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PREFACE

I am thankful to Fadoua Chigr (New Jersey City University) for a diligent perusal of portions of the book. Finally, I prize suggestions on the presentation made by Zbigniew Lipecki (Mathematical Institute, Polish Academy of Sciences).

Szymon Dolecki Tavira October 2023

List of Chapters Chapter I.

Preliminaries

1

Chapter II.

From convergence of sequences to the concept of filter

17

Chapter III.

Convergence of filters

35

Chapter IV.

Continuity

56

Chapter V.

Families of sets

88

Chapter VI.

Pretopologies

107

Chapter VII.

Topological structures

128

Chapter VIII.

Adherences, covers, and compactness

149

Chapter IX.

Topological concepts

162

Chapter X.

Functional study of topologies

189

Chapter XI.

Functional partitions and metrization

236

Chapter XII.

Compact topologies

252

Chapter XIII.

Connected and disconnected topologies

289

Chapter XIV.

Extensions and compactifications

316

Chapter XV.

Uniform structures

334

Chapter XVI.

Sequentially founded convergences

351

Chapter XVII.

Structural aspects

365

Chapter XVIII.

Fundamental classes

381

Chapter XIX.

Diagonality and regularity

419

Chapter XX.

Compactness

437

Chapter XXI.

Mixed properties

479

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xiv

LIST OF CHAPTERS

Chapter XXII.

Implementations and refinements

517

Chapter XXIII.

Completeness

556

Chapter XXIV.

Spaces of maps

582

Chapter XXV.

Duality

608

Chapter XXVI.

Modified duality

624

Chapter XXVII.

Outline of principles

646

Chapter XXVIII. Category theoretic perspective

669

Table of Contents v

About the Author Preface

vii

Chapter I. Preliminaries 1. Relations and maps 2. Equivalence relations 3. Order 4. Polarities 5. Finite and infinite cardinals

1 1 5 6 11 12

Chapter II. From convergence of sequences to the concept of filter 1. Convergence of sequences 2. The concept of filter 3. Order on filters 4. Decomposition into free and principal filters 5. Supplement

17 17 20 26 30 32

Chapter III. Convergence of filters 1. The concept of limit 2. A pointwise perspective 3. Order on convergences 4. Prime convergences 5. Finite convergences 6. Metrizable convergences 7. Supplement

35 35 40 41 45 47 49 53

Chapter IV. Continuity 1. Images and preimages 2. Continuous maps 3. Continuity and order 4. Initial convergences 5. Selections and products 6. Product convergences 7. Diagonal product maps

56 56 58 61 67 69 72 75 xv

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TABLE OF CONTENTS

8. Final convergences 9. *Lower semicontinuity 10. Supplement

77 79 81

Chapter V. Families of sets 1. Isotone and antitone families 2. Ideals, filters, distributive families 3. The notion of grill 4. Ultrafilters 5. Various classes of filters 6. Supplement

88 90 91 94 96 99 103

Chapter VI. Pretopologies 1. Basic facts about pretopologies 2. Adherence 3. Inherence 4. Continuity 5. Structural aspects of pretopologies 6. Pretopologizer 7. Initial density of the Bourdaud pretopology 8. Supplement

107 107 110 113 114 116 119 123 125

Chapter VII. Topological structures 1. Closed sets, closure 2. Open sets, interior, neighborhoods 3. Topologies 4. Structure of the class of topologies 5. Initial density of the Sierpiński topology 6. *Remarks on convergence of nets 7. Supplement

128 128 130 133 137 141 143 145

Chapter VIII. Adherences, covers, and compactness 1. Adherences 2. Covers and inherences 3. Compact convergences 4. Supplement

149 149 152 156 160

Chapter IX. Topological concepts 1. Topological separation 2. Bases 3. Compact, countably compact, and Lindelöf topologies 4. Open, closed, topologically quotient, and perfect maps 5. Supplement

162 162 170 174 179 183

TABLE OF CONTENTS

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Chapter X. Functional study of topologies 1. The real line 2. Real-valued functions 3. More on metrics and metrizable topologies 4. Completely metrizable topologies 5. Completion 6. Functionally closed and open sets 7. Functional separation 8. *A non-trivial regular topology with each continuous function constant 9. Functionally initial topologies 10. Supplement

189 189 191 193 198 203 207 210

Chapter XI. Functional partitions and metrization 1. Infinite sums of positive real numbers 2. Partitions of unity 3. Paracompactness 4. Metrization theorems 5. Supplement

236 236 236 242 245 248

Chapter XII. Compact topologies 1. A short story of compactness 2. Compact sets in topological spaces 3. Weaker variants of compactness 4. Sequentially compact topologies 5. The Cantor set 6. *Cardinality aspects 7. The Stone topology 8. *Almost disjoint families 9. *Hyperconvergences and semicontinuities 10. Supplement

252 252 257 258 261 263 267 269 276 281 285

Chapter XIII. Connected and disconnected topologies 1. Connected topologies 2. Components and quasi-components 3. Locally and arcwise connected topologies 4. Disconnected topologies 5. Ultrametric and ultrametrizable spaces 6. Cantor-Bendixon theorem 7. Supplement

289 289 293 295 300 301 307 310

Chapter XIV. Extensions and compactifications 1. Extensions of spaces and maps

316 316

219 224 229

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2. 3. 4. 5. 6.

Compactifications Filters based in lattices Čech-Stone compactification A construction of maximal compactifications Supplement

319 322 325 330 332

Chapter XV. Uniform structures 1. Preuniformities 2. Uniformities 3. Operations 4. Completeness 5. Supplement

334 334 341 344 346 350

Chapter XVI. Sequentially founded convergences 1. Sequential filters 2. Subsequences and subquences 3. Sequentially founded convergences 4. *Extrema of sets of sequential filters 5. Supplement

351 351 353 355 359 363

Chapter XVII. Structural aspects 1. Projective and coprojective classes 2. Functors 3. Reflective and coreflective classes 4. Initial and final functors 5. Supplement

365 365 369 371 374 379

Chapter XVIII. Fundamental classes 1. Stability and pavements 2. Foundations 3. Pseudotopologies 4. *Initial density, exponential hulls 5. Adherence-determined convergences 6. Paratopologies 7. Hypotopologies 8. Commutation of functors with products 9. Classification of quotient maps 10. Supplement

381 381 386 390 395 396 400 406 410 412 417

Chapter XIX. Diagonality and regularity 1. Contour operation 2. Diagonality 3. Regularity 4. *Partial regularization

419 419 423 428 432

TABLE OF CONTENTS

5. *P-points 6. Supplement

xix

433 436

Chapter XX. Compactness 1. Compactness versus cover-compactness 2. From pseudotopologies to compactness 3. Conditional compactness 4. Countable and sequential compactness 5. Lindelöf property 6. Compactoid and compact families 7. Conditional compactness of products 8. Regularity and topologicity in compact pseudotopologies 9. Local compactness 10. Hyperconvergences revisited 11. Classification of perfect maps 12. Compactness related properties of maps 13. Supplement

437 437 439 444 446 449 449 454

Chapter XXI. Mixed properties 1. Fréchet pretopologies 2. Galois connections 3. Functorial equations and inequalities 4. Sequentially induced convergences 5. Sequentially founded convergences induced by adherence-determined classes 6. Subclasses induced by locally compact convergences 7. Preservation of mixed properties under quotient maps 8. Preservation of mixed properties under products 9. *Relations and polarities in classes of filters 10. Coreflectors for mixed properties 11. Supplement

479 480 483 485 488

Chapter XXII. Implementations and refinements 1. Active boundary 2. *Topological defect and sequential order 3. *Sequential cascades 4. Subsequential topologies 5. Topologically maximal pretopologies 6. Applications of the Stone topology 7. *Homogeneity quest

517 517 521 526 533 535 536 539

458 459 465 468 473 476

493 496 498 501 505 509 514

xx

TABLE OF CONTENTS

8. *A compact Fréchet topology with non-Fréchet square 9. Measure-theoretic convergences

543 545

Chapter XXIII. Completeness 1. Fundamental filters and completeness 2. Cocompleteness 3. Finitely complete convergences 4. Countably complete convergences 5. Completeness number 6. Completeness of subspaces 7. Operations preserving completeness number 8. Baire category theorem 9. *Ultracompleteness 10. Supplement

556 556 558 560 564 568 570 573 575 577 581

Chapter XXIV. Spaces of maps 1. Dual (natural) convergence 2. Exponentiation, transposition, and adjoint maps 3. Exponential law 4. Hyperconvergences as dual convergences 5. Other convergences on function spaces 6. A unification of functional convergences and hyperconvergences 7. Consonance 8. Compactness in function spaces

582 583 586 591 593 595 601 602 604

Chapter XXV. Duality 1. Dual and bidual convergences 2. Bidual reflector 3. Reflective inequalities in duality theory 4. Exponential classes and hulls 5. Functionally embedded convergences 6. Epitopologies 7. Duality in completeness

608 608 611 613 615 618 620 621

Chapter XXVI. Modified duality 1. Functorial modifications of duality 2. Applications to mixed properties 3. Exponential objects 4. Topologicity of dual convergences 5. Conditional versions of core compactness 6. Strongly and productively sequential convergences

624 624 629 631 635 637 641

TABLE OF CONTENTS

xxi

Chapter XXVII. Outline of principles 1. Axioms of set theory 2. Cardinality 3. Well-ordered sets 4. Ordinal numbers 5. Ordinal arithmetic 6. Cardinal ordinals (Alephs) 7. Closed unbounded sets 8. Well-founded relations 9. Small cardinals 10. Supplement

646 647 649 652 654 659 659 661 662 663 667

Chapter XXVIII. Category theoretic perspective 1. Objects, morphisms, and functors 2. Constructs 3. Topological categories 4. Reflections and coreflections in concrete categories 5. Conclusion

669 670 672 675 679 681

Postface

682

Bibliography

684

List of Symbols

696

Index

701

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CHAPTER I

Preliminaries While the ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) axioms are assumed throughout this book, the Preliminaries are addressed first and foremost to a reader with only basic set-theoretic background. An advanced reader may have a look at it just to get acquainted with notation and terminology. Axiomatic considerations are gathered in Chapter XXVII. I.1. Relations and maps In this preliminary section, we shall gather a few basic facts about relations and maps. Mind that we distinguish between the symbol = of equality or equation, and the symbol := of definition. For instance, in 3 · 13 = 1, the said symbol denotes an equality, in the expression x + x2 = 1, where x is a complex number, an equation, which becomes an equality, true or false, when we replace x with a complex number. When we write 0! := 1, and (n + 1)! := (n + 1) · n! for n > 0,

we inductively define the factorial n! for each natural number n. In mathematics, the notion of set is not defined. We use it intuitively, but its sense is made precise by a series of rules (called axioms) that are presented in Chapter XXVII. We say that x is an element of a set X, or x belongs to X, in symbols, x ∈ X. This relation is also undefined, and its meaning can be grasped thanks to the mentioned axioms. We define, however, a subset A of a set X, as follows: A ⊂ X ⇐⇒ (x ∈ A =⇒ x ∈ X)

for each x. The set-theoretic difference X \ A is defined by x ∈ X \ A ⇐⇒ (x ∈ X) ∧ (x ∈ / A).

Let us yet mention that a mathematical definition should be based, in one way or another, on the notions of set and element (see Chapter XXVII for details). 1

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latexit


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latexit

RA
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latexit


aj0 . R be the real line with the standard order ≤. ThenWfor each A ⊂ R, which is bounded above, which means that A+ 6= ∅, there exists A. If we consider, the subset Q of R, consisting of rational numbers, then A := {q ∈ Q : q 2 < 2}, then

10Let

I.3. ORDER

9

There exist alternative notations for extrema (that is, suprema and infima), especially in the case of the ordered real line (R, ≤), namely _ ^ sup A = A, inf A = A. W Definition I.3.6. An ordered set (X, ≤) is called a lattice if A V and A exist for each non-empty finite subset A of X; a complete lattice whenever they exist for each ∅ 6= A ⊂ X. Definition I.3.7. A bijective order-preserving map f is said to be an order isomorphism if the inverse map f −1 is order-preserving. It follows directly from the definitions that Proposition I.3.8. An order isomorphism f preserves the extrema, that is, _ _ ^ ^ f ( A) = f (A), f ( A) = f (A).

Example I.3.9. The ordered extended real line (R, ≤), where R := R ∪ {−∞, +∞} is a complete lattice, while the ordered real line (R, ≤) is only a relatively complete lattice, that is, each non-empty bounded set has a supremum and an infimum (11). The ordered set of Example I.3.1 is a lattice, which is necessarily complete, because X is finite. More generally, for each set X, the set L = 2X of subsets of X is a complete lattice with respect to the inclusion ⊂, where A ⊂ B ⇐⇒ (x ∈ A ⇒ x ∈ B). W X for A, B V ∈ 2 . Indeed, for any A ⊂ 2X , the supremum A and the infimum A are given by _ [ [ A, A= A := A∈A ^ \ \ A= A := A. A∈A

If (L, ≤) is an ordered set, and M is a subset of L, then (M, ≤) constitutes an ordered set with the restriction of the relation ≤ to M . In other words, if x, y ∈ M , then x ≤ y with respect to (M, ≤) if and only if x ≤ y with respect to (L, ≤). However, suprema and infima considered in (M, ≤) and in (L, ≤) are, in general, different!

A+ = {q ∈ Q : 2 ≤ q 2 } is non-empty, but does not have a least element (because √ 2∈ / Q), so that the supremum of A does not exist. 11More precisely, non-empty bounded above sets admit suprema, and bounded below sets admit infima.

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latexit


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latexit


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

latexit


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

R A 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

'(n4 )

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

'(n3 )

AAAC0HicjVHLSsNAFD3G97vq0k1QBEUoSV3osujGpYpVwZYyGaft0LycTMQiIm5dC271q8Q/0L/wzhhBLaITkpw5954zc+8N0lBm2vNeBpzBoeGR0bHxicmp6ZnZ0tz8UZbkiosaT8JEnQQsE6GMRU1LHYqTVAkWBaE4Dro7Jn58IVQmk/hQ91LRiFg7li3JmSaqUb9gKu1IdzVu+mvN0rJX9uxy+4FfgOXqUn39/qXa20tKz6jjDAk4ckQQiKEJh2DI6DmFDw8pcQ1cEacISRsXuMYEaXPKEpTBiO3St02704KNaW88M6vmdEpIryKlixXSJJSnCJvTXBvPrbNhf/O+sp7mbj36B4VXRKxGh9i/dJ+Z/9WZWjRa2LI1SKoptYypjhcuue2Kubn7pSpNDilxBp9RXBHmVvnZZ9dqMlu76S2z8VebaViz50VujjdzSxqw/3Oc/eCoUvY3ypV9mvQ2PtYYFrGEVZrnJqrYxR5q5H2OBzziyTlwLp0b5/Yj1RkoNAv4tpy7dziiluE=

'(n1 )

ℵ0 .

Proposition I.5.7. The set R of real numbers is uncountable.

Proof. Consider the set R of real numbers. Suppose that, on the contrary, there exists a surjective map ϕ ∈ RN . By assumption, there exists a first natural number n1 such that ϕ(0) < ϕ(n1 ). Again, by assumption, there exists a first natural number n2 such that ϕ(0) < ϕ(n2 ) < ϕ(n1 ).

Figure I.5.1. Any sequence ϕ cannot fill the real line.

∈ Q admits many fractional representations; for example,

1 3

=

2 6

=

3 9

=

11 33 .

I.5. FINITE AND INFINITE CARDINALS

15

Accordingly, there is a strictly increasing sequence (nk )k such that, for each k ∈ N, ϕ(n2k ) < ϕ(n2k+2 ) < . . . < ϕ(n2k+3 ) < ϕ(n2k+1 ). As each non-empty bounded subset of R has an infimum and a supremum, there exists r ∈ R such that supk∈N ϕ(n2k ) ≤ r ≤ inf k∈N ϕ(n2k+1 ).



By the construction, r ∈ / ϕ(N), which is a contradiction.

The cardinality of R is denoted by c and called continuum. We should be aware from the very beginning that there are infinitely many uncountable cardinalities, because Theorem I.5.8 (Cantor). For every set X, (I.5.2)

card X < card 2X .

Proof. The map h(x) := {x} is an injection from X to 2X , hence card X ≤ card 2X . If f : X → 2X , then we claim that the set {x ∈ X : x∈ / f (x)} does not belong to f (X), that is, f is not surjective. If on the contrary, there were x0 ∈ X such that

(I.5.3)

f (x0 ) = {x ∈ X : x ∈ / f (x)},

then either x0 ∈ f (x0 ), that is, x0 ∈ / f (x0 ) by (I.5.3), or x0 ∈ / f (x0 ), hence x0 ∈ f (x0 ) by (I.5.3). We have got a contradiction in any case. Therefore, no f : X → 2X is a surjection, and hence (I.5.2). 

Consequently, by an easy induction (19), we infer that there are infinitely many infinite cardinals. By definition, λκ = card(Y X ), where λ = card Y and κ = card X. Therefore, 2κ := card 2X = card({0, 1}X ) for any set X such that card X = κ. Indeed, there exists a bijection between maps f ∈ {0, 1}X and subsets A ⊂ X, realized with the aid of characteristic functions χA ( 1, x ∈ A, (I.5.4) χA (x) := 0, x ∈ / A. In particular, χX (x) = 1 for each x ∈ X, and χ∅ (x) = 0 for each x ∈ X. 19Starting

X

from any set X, we get card X < card 2X < card 2(2

)

, and so on.

16

I. PRELIMINARIES

There exists a widespread convention to denote by [X] 0, there exists nε ∈ N such that |ϕ(n) − x| ≤ ε for every n ≥ nε .

Let us reformulate this basic definition in a way that will reveal a novel aspect, which is fundamental for our approach. A strict interval joining a and b will be denoted by (1) ]a, b[ := {r ∈ R : a < r < b}.

As |ϕ(n) − x| < ε amounts to ϕ(n) ∈ ]x − ε, x + ε[ , we infer that

Definition (B). A sequence ϕ on the real line converges to x if and only if {n ∈ N : ϕ(n) ∈ / ]x − ε, x + ε[} is finite for each ε > 0. 1We

adopt here the notation ]a, b[ rather than (a, b), because the latter symbol denotes an ordered pair. This is a traditional French notation. 17

18

II. FROM CONVERGENCE OF SEQUENCES TO THE CONCEPT

Let us check that the two definitions are equivalent. Let ε > 0. If |xn − x| ≤ ε holds for n ≥ nε , then {n : |xn − x| > ε} is finite. Conversely, if {n : |xn − x| > ε} is finite, then letting nε := max{n : |xn − x| > ε} + 1, we assure that |xn − x| ≤ ε for each n ≥ nε . A subset A of a set N is called cofinite if N \A is finite. Accordingly, Definition (B) states that a sequence ϕ converges to x whenever the set {n ∈ N : ϕ(n) ∈ ]x − ε, x + ε[}

is cofinite for each ε > 0. Notice that Definition (B) does not mention the standard order on the set N of indices of a sequence, which was used in the standard Definition (A)! It follows, in particular, that any permutation of indices of a sequence does not affect its convergence. Therefore, the order on the set of indices turns out to be useless, or at least, unnecessary from the convergence point of view. Therefore, instead of using N for the set of indices, we can take an arbitrary infinite countable set N (2). This leads to

Definition (C). A sequence ϕ ∈ RN converges to x if and only if, for each ε > 0, there exists a cofinite subset F of N such that ϕ(F ) ⊂ ]x − ε, x + ε[.

We notice that if F is a cofinite subset of N and F ⊂ G ⊂ N , then G is also a cofinite subset of N . Any finite intersection of cofinite subsets of N is cofinite (because any finite union of finite sets is finite). Finally, the empty subset of N is not cofinite, because N is infinite. An interval ]x − ε, x + ε[ is a particular (symmetric) neighborhood of x. More generally, a subset V of R is called a neighborhood of x if there exist numbers a and b such that a < x < b, and ]a, b[ ⊂ V . Denote by N (x) the family of all neighborhoods of x. Notice that (isotone)

(finitely complete) (proper)

W ⊃ V ∈ N (x) =⇒ W ∈ N (x),

V0 , V1 ∈ N (x) =⇒ V0 ∩ V1 ∈ N (x), ∅∈ / N (x).

Actually, the formula (proper) is a consequence of a stronger property of neighborhoods, that is, x ∈ V for each V ∈ N (x). Let N be an infinite countable set and ϕ ∈ X N . Let Γϕ be the family of subsets of X defined by (II.1.1) 2A

E ∈ Γϕ ⇐⇒ ϕ− (E) is cofinite in N.

map from an infinite countable set N to a set X is called a quence on X [83].

II.1. CONVERGENCE OF SEQUENCES

19

In other words, E ∈ Γϕ if and only if there exists a cofinite subset A of N such that ϕ(A) ⊂ E. Notice that (isotone)

(finitely complete) (proper)

F ⊃ E ∈ Γϕ =⇒ F ∈ Γϕ,

E0 , E1 ∈ Γϕ =⇒ E0 ∩ E1 ∈ Γϕ, ∅∈ / Γϕ.

Proposition II.1.1. Let ϕ be a sequence and Γϕ be defined by (II.1.1). Then ϕ converges to x if and only if N (x) ⊂ Γϕ.

Proof. In fact, suppose that ϕ converges to x, and let V ∈ N (x). By the definition of N (x), there exists ε > 0 such that ]x−ε, x+ε[ ⊂ V . By definition of Γϕ, there is E ∈ Γϕ such that E ⊂ V , hence by (isotone), V ∈ Γϕ. Conversely, if ]x − ε, x + ε[ ∈ Γϕ, then {n : ϕ(n) ∈ / ]x − ε, x + ε[ } is finite, hence ϕ converges to x.  Subsequences. A sequence (yk )k is called a subsequence of a sequence (xn )n whenever there exists a strictly increasing (nk )k such that yk = xnk for each k < ω. In other terms, a sequence ψ is a subsequence of a sequence ϕ if there exists a strictly increasing map γ : N −→ N such that ψ(k) = ϕ(γ(k)) for each k ∈ N. As you expect, if ψ is a subsequence of a sequence ϕ, then (II.1.2)

Γϕ ⊂ Γψ.

Indeed, E ∈ Γϕ if and only if ϕ− (E) is cofinite. As γ is strictly increasing, (3) ψ − (E) = γ − (ϕ− (E)) is cofinite, hence E ∈ Γψ. Hence, Corollary II.1.2. If a sequence ϕ converges to x and ψ is a subsequence of ϕ, then ψ converges to x. Of course, we do not need the order on indices to define a subsequence. Proposition II.1.3. If ϕ ∈ X N and ψ ∈ X K , where N and K are infinite countable sets, then ψ is a subsequence of ϕ if and only if there is an injection γ ∈ N K such that ψ = ϕ ◦ γ. It follows that

Lemma II.1.4. Γϕ ⊂ Γψ if and only if, for each cofinite subset G of N , the set ψ − (ϕ(G)) is cofinite in K. 3Then

there is k0 such that {γ(k) : k > k0 } ⊂ ϕ− (E).

20

II. FROM CONVERGENCE OF SEQUENCES TO THE CONCEPT

In particular, if ψ is a subsequence of ϕ, then the condition of Lemma II.1.4 holds. More generally, if ψ = ϕ ◦ γ and γ − (G) is cofinite for each cofinite subset G of N , then Γϕ ⊂ Γψ (4). From this perspective, ϕ ⊂ N × X and ψ ⊂ K × X, so that ψ − ⊂ X × K is the inverse relation of ψ. The composed relation ψ − ◦ ϕ ⊂ N × K, that we call the correlation ϕ with ψ, can be written explicitly (II.1.3)

ψ − ◦ ϕ = {(n, k) : ϕ(n) = ψ(k)}.

Now, Lemma II.1.4 can be rephrased: Γϕ ⊂ Γψ if and only if the correlation ψ − ◦ ϕ preserves cofinite sets. The accomplishment of this section is twofold: we have shown that, from the convergence point of view, a sequence ϕ : N −→ X can be, not only freed from the order on the set N of its indices, but ultimately, relieved from the set of indices altogether. This looks like a miracle, but it is merely a pertinent perspective. Not everything, however, is abandoned during this change of viewpoint, because the structure of cofinite sets is a residue of the natural order on the natural numbers. On the other hand, it is often instrumental to deploy natural numbers in construction or description of sequences. II.2. The concept of filter In the preceding section we have observed that the convergence of a sequence to a point of the real line can be characterized by the inclusion between two families of sets, one related to the sequence, the other related to the point. Moreover, these families have some common properties, that is, they are (isotone), (finitely complete), and (proper). Let X be a non-empty set. A family F of subsets of X is said to be isotone if (isotone)

F ∈ F and F ⊂ G =⇒ G ∈ F.

A family F of subsets of X is called a filter (on X) if it is (isotone) and (finitely complete)

F0 ∈ F and F1 ∈ F =⇒ F0 ∩ F1 ∈ F.

A filter F is said to be proper if (proper)

∅∈ / F.

4As we shall see in Section XVI.2, this sufficient condition will become also necessary

if we allow γ to be defined on a cofinite subset of K.

II.2. THE CONCEPT OF FILTER

21

Non-empty proper filters are called non-degenerate. If the condition (proper) does not hold, then we say that F is improper (5), degenerate if it is either improper or empty. The family of neighborhoods of a point, and the family associated with a sequence by (II.1.1) are proper filters. We denote by F the class of non-empty proper filters, and by F the class of all filters. If X is a set, then the set of all non-empty proper filters on X is denoted by FX (6), and of all filters on X by FX. The only improper filter on a (non-empty) set X is 2X (the family of all subsets of X), because by (isotone), ∅ ∈ F implies that G ∈ F for each ∅ ⊂ G ⊂ X. Hence FX = FX ∪ {∅, 2X }. Definition II.2.1 (Filter-base). A subfamily B of a proper filter F is called a base (or a filter-base) of F if for each F ∈ F, there exists B ∈ B such that B ⊂ F . We say also that B generates F. In particular, each filter constitutes its own filter-base. Proposition II.2.2. A non-empty family B of subsets of X is a filter-base if and only if ∅ ∈ / B and for each B0 , B1 ∈ B, there exists B ∈ B such that B ⊂ B0 ∩ B1 . Proof. If F is a proper filter and B ⊂ F, then B0 ∩ B1 ∈ F. If B is a filter-base of F, then there exists B ∈ F such that B ⊂ B0 ∩ B1 . Conversely, if the condition holds, then the family F such that, for each F ∈ F there exists B ∈ B with B ⊂ F , is a filter. Obviously, F is isotone. If now F0 , F1 ∈ F, then there exist B0 , B1 ∈ B such that B0 ⊂ F0 and B1 ⊂ F1 , hence there exists B ∈ B such that B ⊂ B0 ∩ B1 ⊂ F0 ∩ F1 , hence F0 ∩ F1 ∈ F. Moreover, ∅ ∈ / F for otherwise there would be B ∈ B such that B ⊂ ∅.  If A ⊂ 2X , then

(II.2.1)

A↑ := {F ⊂ X : ∃ F ⊃ A}. A∈A

Sure enough, a family A is isotone if and only if A = A↑ . It is plain that B is a filter-base of F if and only if B ↑ = F. If A = {{x}}, then we abridge x↑ := {{x}}↑ . 5The

empty family of subsets of a non-empty set is a proper filter. make a distinction between a class (of proper filters) and a set (of filters on a given set). This is because, by the axioms of set theory (Chapter XXVII), all proper filters do not constitute a set, but the proper filters on any set X do. 6We

22

II. FROM CONVERGENCE OF SEQUENCES TO THE CONCEPT

Definition II.2.3 (Filter-subbase). A family D of subsets of X is called centered or a filter-subbase if \ (II.2.2) D0 ⊂fin D =⇒ D 6= ∅. D∈D0

(II.2.2) is called the finite intersection property. T If D is a subbase, then { D0 : D0 ⊂fin D} is a filter-base.

Definition II.2.4 (Principal filter). A filter is called principal (or finitely based, or else of finite character ) if it has a finite filter-base. Consequently, each principal filter on X has a filter-base of the form {B}, where B ⊂ X, because {B}↑ := {F ⊂ X : B ⊂ F } in view of (II.2.1). Of course, {B}↑ is proper if and only if B 6= ∅. The class of all proper principal filters T is denoted by F0 . Notice that if a proper filter F is principal, then F ∈F F ∈ F. Let us extend the concept of cofinite set introduced in Section II.1. Definition II.2.5 (Cofinite set). Let X be a set. A subset F of X is said to be cofinite (in X) if X \ F is finite. The family of all cofinite subsets of X is denoted by (X)0 .

Proposition II.2.6. The family (X)0 of cofinite subsets of a nonempty set X is a proper filter if and only if X is infinite. Proof. Indeed, if F0 , F1 are cofinite subsets of X, then X \ (F0 ∩ F1 ) = (X \ F0 ) ∪ (X \ F1 ) is finite, hence F0 ∩ F1 is cofinite. Now, X is finite if and only if ∅ ∈ (X)0 , equivalently (X)0 is improper.  T Definition II.2.7 (Free filter). A filter F is called free if F ∈F F = ∅.

Lemma II.2.8. A proper filter F on an infinite set X is free if and only if F ⊃ (X)0 . T Proof. By definition, a filter F on X is free if and only if F ∈F F = ∅, that is, if and only if for each x ∈ X, there is F ∈ F such that x∈ / F , hence X \ {x} ∈ F. By Formula (finitely complete), finite intersections of such sets also belong to F, that is, X \ G ∈ F for each finite subset G of X. In other words, a free filter on X contains all cofinite T subsets of X. Conversely, H∈(X)0 H = ∅, because X \ {x} ∈ (X)0 for T every x ∈ X. Hence, if F ⊃ (X)0 then F ∈F F = ∅. 

Definition II.2.9 (Sequential filter). A filter F on a set X is called sequential if there exists a sequence ϕ ∈ X N such that F = Γϕ (II.1.1).

II.2. THE CONCEPT OF FILTER

23

The class of all sequential filters is denoted by S, and the set of all sequential filters on X by SX or εX. The same definition can be rewritten traditionally, which reads as follows Proposition II.2.10. A filter F on a set X is sequential if and only if there exists a sequence (xn )n of elements of X so that (II.2.3) is a filter-base of F.

{{xk : k ≥ n} : n ∈ N}

Remark II.2.11. Recall that traditionally a sequence (xn )n on X is indexed by the set of natural numbers N, and it is identified with a map ϕ ∈ X N to the effect that ϕ(n) := xn . Moreover, we have pointed out that the order of N is irrelevant for convergence questions concerning sequences, so that it is enough to view sequences on X as maps ϕ ∈ X N , where N is any infinite countable set. Corollary II.2.12. Each sequential filter contains a countable set.

We say that a sequence ϕ is free if Γϕ is a free filter, principal if Γϕ is a principal filter. By definition, each sequential filter on a set X is of the form Γϕ, where ϕ ∈ X N and N is an infinite countable set.

Proposition II.2.13. A filter Γϕ is free if and only if ϕ− (x) is finite for each x ∈ ϕ(N ).

Proof. Suppose that there is x ∈ ϕ(N \A) for each A ⊂fin N , that is, ϕ− (x)\A 6= ∅ for each A ⊂fin N , hence ϕ− (x) is infinite. Conversely, if for every x ∈ ϕ(N ) there is A ⊂fin N such that x ∈ / ϕ(N \ A), then ϕ− (x) ⊂ A, thus ϕ− (x) is finite. 

Proposition II.2.14. A filter Γϕ is principal if and only if ϕ− (x) is infinite for each x in a cofinite subset of ϕ(N ).

Proof. Let E be a non-empty subset of X such that Γϕ = {E}↑ . It means that E ⊂ ϕ(F ) for each cofinite subset F of N . The set D := {x ∈ X : card ϕ− (x) < ∞} is disjoint from E, because E ⊂ ϕ(N \ ϕ− (x)) for each x ∈ D. Accordingly, if x ∈ E then ϕ− (x) is infinite. On the other hand, D is finite, because otherwise ϕ− (D) would be an infinite subset of N such that ϕ− (D) ∩ F 6= ∅ for each cofinite subset F of N . Accordingly, ψ := ϕ|ϕ− (D) would be a subsequence of ϕ, which is impossible, because ψ(ϕ− (D)) ∩ E = ∅. The filter Γϕ has a countable base of the form {ϕ(N \ A) : A ⊂fin N }. If Γϕ is not principal, then the filter-base is infinite, hence there exists a strictly decreasing base {Bk : k < ω}. If xk ∈ Bk \ Bk+1 , then

24

II. FROM CONVERGENCE OF SEQUENCES TO THE CONCEPT

T (xk )k is a free subsequence of ϕ, and {xk : k < ω} ∩ 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

m} = A for each m < ω. If A is infinite countable, say A = {ak : k ∈ N}, then consider a sequence ϕ defined by a0 ,

a0 , a1 ,

a0 , a1 , a2 ,

a0 , a1 , a2 , a3 ,

...,

and thus {ϕ(n) : n > m} = A for each m < ω.



Example II.2.23 (Countably based, non-sequential filter). Let x ∈ R and consider the family of intervals of the real line  B := ]x − n1 , x + n1 [ : n ∈ N1 .

This family is evidently countable, but each non-empty strict interval of the real line is uncountable (9). Therefore the filter B ↑ is not sequential.

Example II.2.24 (Cocountable filter). We say that a subset F of X is cocountable if X \ F is a countable set or empty; the family of all cocountable subsets of X is denoted by (X)1 . More generally, if B ⊂ X, then a subset F of X is called cocountable subset of B (or B-cocountable) if B \F is countable or empty. The family (B)1 of all Bcocountable sets is a filter, called the cocountable filter at B; it is proper if and only if B is uncountable, because then B \ ∅ is not countable, and thus ∅ ∈ / (B)1 . Each cocountable filter of an uncountable set is free, because (B)0 ⊂ (B)1 . II.3. Order on filters It is easy to see that the inclusion ⊂ on the set FX is reflexive, antisymmetric, and transitive. Therefore, (FX, ⊂) is an ordered set. It follows from definitions that if F0 , F1 ∈ FX, then (II.3.1)

(II.3.2)

F0 ∨ F1 = {F0 ∩ F1 : F0 ∈ F0 , F1 ∈ F1 } , F0 ∧ F 1 = F0 ∩ F 1 .

If F0 , F1 are proper filters, then F0 ∧F1 is proper, but F0 ∨F1 is proper only if F0 ∩ F1 6= ∅ for each F0 ∈ F0 and F1 ∈ F1 .

Example II.3.1. If B1 , . . . , Bn are countable, infinite sets, then (B0 )0 , . . . , (Bn )0 are free cofinite filters. Then the infimum (B0 )0 ∧ . . . ∧ (Bn )0 = (B1 ∪ . . . ∪ Bn )0 . The supremum (B0 )0 ∨ . . . ∨ (Bn )0 = (B1 ∩ . . . ∩ Bn )0 is proper whenever B1 ∩ . . . ∩ Bn is infinite.

9Recall

that the real line R is uncountable. Every non-empty strict interval of R is uncountable as a bijective image of R.

II.3. ORDER ON FILTERS

27

Recall that an ordered set, in which a supremum and an infimum exist for any two elements, hence for any finite set of elements, is called a lattice; a complete lattice whenever all non-empty subsets admit suprema and infima (Definition I.3.3). If A is a family of subsets of X, then \ A∩ := { B : B ⊂fin A} denotes the family of finite intersections of the elements of A.

Proposition II.3.2. The set of all filters on a given set, ordered by inclusion, is a complete lattice; if H is a non-empty set of (possibly improper) filters on X, then ^ \ _ [ H= H, H=( H)∩ . H∈H H∈H T Proof. The family H∈H H isTincluded in every filter from H, and it is a filter. Indeed, if F0 , F1 ∈ H∈H H, then F0 , F1 ∈ HTfor each H ∈ H, hence F0 ∩FS 1 ∈ H for each H ∈ H, so that F0 ∩F1 ∈ H∈H H. On the other hand, H∈H H is the least family of sets including every filter from H, but in general it is not a filter. We get the least (possibly improper) filter S that includes it, by taking all the finite intersections of the elements of H∈H H.  W Corollary II.3.3. T The supremum H of a set HS of filters is proper if and only if G 6= ∅ for each finite family G ⊂ H∈H H.

Consequently, the set FX of proper filters is not even a lattice. Although the intersection of each non-empty set H ofS proper filters S is a proper filter, the supremum exists in FX if and only if H := H∈H H is closed under finite intersections. Otherwise, the supremum exists in FX = FX ∪ {2X }, and is equal to the degenerate filter 2X . Proposition II.3.4. V S Each infimum of principal filters is a principal filter: A∈A {A}↑ = { A∈A A}↑ . However,

Proposition II.3.5. Each filter is a supremum of principal filters. W Proof. Let F be an arbitrary filter. Then F = F ∈F {F }↑ . Of course, On the other hand, if F0 ∈ F then F0 ∈ F0↑ , hence W ⊃ holds. F0 ∈ F ∈F {F }↑ .  Proposition II.3.6. Each infimum of free filters is free.

Proof. Indeed, by Lemma II.2.8, a filter F on X is free V if and only if F ⊃V (X)0 , hence if F is a set of free filters on X, then F ⊃ (X)0 , hence F is free. 

28

II. FROM CONVERGENCE OF SEQUENCES TO THE CONCEPT

Proposition II.3.7. Each countably based filter is an infimum of sequential filters. Proof. Let F be a countably based filter on a set X, then {E ∈ S : E ⊃ F} 6= ∅. Indeed, let (Fn )n be a decreasing filter-base of F. If there exists n0 such that Fn = Fn0 for each n ≥ n0 then there exists x ∈ Fn for each n < ω. If ϕ(n) := x, then Γϕ = x↑ ⊃ F. Otherwise, there exists a strictly increasing sequence (nk )k and T xk ∈ Fnk \ Fnk +1 . If now ϕ(k) := xk , then Γϕ ⊃ F. Of course, F ⊂ {E ∈ S : E ⊃ F}. Conversely, if H ∈ / F and (Fn )n is a filter-base of F, then Fn \H 6= ∅ for each n < ω. Accordingly, there exists a sequence (xn )n T such that xn ∈ Fn \ H. Let ϕ(n) :=Txn . Then Γϕ ⊃ F and H ∈ / Γϕ ⊃ {E ∈ S : E ⊃ F}. Therefore, F ⊃ {E ∈ S : E ⊃ F}.  If F is a countably carried filter, then there exists a sequential filter E ⊂ F. Indeed, a countable set E ∈ F, then {E}↑ ⊂ F. The class E is isotone, that is, if E ∈ E and F ⊃ E, then F ∈ E. The class S of sequential filters is not isotone, because by Proposition V.4.8, there exist non-sequential filters finer than sequential filters. Definition II.3.8 (Fréchet filter). A filter is called Fréchet if it is an infimum of sequential filters. Let us warn that the term Fréchet filters has been often employed to designate sequential filters in our sense (10)! Proposition II.3.7 implies that every countably based filter is Fréchet. As we shall see in Example XVI.4.5, there exist Fréchet filters which are not countably based. If F is a filter on X and A ⊂ X, then we abridge (II.3.3)

F ∨ A := F ∨ {A}↑ ,

and call F ∨ A the restriction of F to A. Notice that F ∨ A is proper if and only if A ∩ F 6= ∅ for each F ∈ F. It is also handy to use another abbreviation, namely (II.3.4)

F \ A := F ∨ (X \ A)↑ .

Definition II.3.9 (Ultrafilter). A proper filter U is called an ultrafilter if H ⊃ U implies H = U for every proper filter H.

10We

have adopted the current terminology for several reasons. The main reason is that there is a class of topological spaces traditionally called Fréchet spaces. They can be characterized as the spaces, every vicinity filter of which is Fréchet in the sense of Definition II.3.8. Moreover, it is preferable to give a mathematical object a common name evoking its characteristic features, rather than that of a mathematician who contributed to its study. In particular, sequential filters are so common and were effectively studied by many a mathematician.

II.3. ORDER ON FILTERS

29

We denote by U the class of all ultrafilters. For each proper filter F, there exists an ultrafilter U greater than F (11). The set of all ultrafilters which include a filter F is denoted by βF. Observe that if F and G are filters, then F ⊂ G implies βF ⊃ βG. Consequently, the set of all ultrafilters on X is denoted by UX or by βX (as a shorthand for β{X}). In other words, ultrafilters on X are maximal elements of FX. If X is a non-empty set, then for any x ∈ X, x↑ := {F ⊂ X : x ∈ F }

is an ultrafilter. Indeed, if H is a filter on X such that x↑ ⊂ H and H ∈ H \ x↑ , then x ∈ / H and thus H ∩ {x} = ∅, contradicting Formula (finitely complete). In view of (II.2.1), x↑ = {{x}}↑ . We shall examine ultrafilters in detail in Section V.4. Let us observe right now that 2X UX FX

{X}

Figure II.3.1. The coarsest filter on X is {X}, the UX consists of maximal proper filters, and the degenerate filter 2X is the finest. Proposition II.3.10. A filter F on X is an ultrafilter if and only if either A ∈ F or X \ A ∈ F for every ∅ 6= A ⊂ X.

Proof. If F is an ultrafilter on X and ∅ 6= A ∈ / F, then F \A 6= ∅ for each F ∈ F in view of Formula (isotone). Hence {F \ A : F ∈ F} is a filter-base, so that F \ A is a proper filter. Evidently, F \ A ⊃ F, and thus X \ A ∈ F \ A = F. Conversely, if there exists ∅ 6= A ∈ / F so that A ∈ / F and X \A ∈ / F, then F \ A ⊃ F and F ∨ A ⊃ F. These filters are proper and distinct 11By

Corollary XXVII.4.9 of the Kuratowski-Zorn Theorem XXVII.4.8.

30

II. FROM CONVERGENCE OF SEQUENCES TO THE CONCEPT

from F (because A ∈ F ∨ A and X \ A ∈ F \ A), hence F is not an ultrafilter.  II.4. Decomposition into free and principal filters The kernel of a family A of subsets of a set X is defined as the intersection of its elements: \ ker A := A. A∈A

Therefore, a filter F is free if and only if ker F = ∅. Hence, a filter F is principal if and only if ker F ∈ F. The class of free filters is denoted by F∗ and the set of free filters on a set X by F∗ X. For a filter F on X, the family (II.4.1)

F ∗ := F \ ker F

is a free (possibly improper) filter, called the free part of F. Notice that a filter F is free if and only if F ∗ = F. On the other hand, a filter F is principal if and only if F ∗ is improper. Theorem II.4.1 (Filter decomposition). For every filter F on X, there exists a unique pair (F ∗ , F • ) of (possibly improper) filters such that F ∗ is free, F • is principal, and F = F ∗ ∧ F • and F ∗ ∨ F • = 2X .

Proof. If F is a filter on a set X, then let F • := (ker F)↑ and F := F \ker F, as in (II.4.1). By definition, ker F ⊂ F for each F ∈ F, hence F ⊂ (ker F)↑ = F • (12). As F \ ker F ⊂ F for every F ∈ F, also F ⊂ F \ ker F = F ∗ . Therefore, F ⊂ F ∗ ∧ F • . On the other hand, if F ∈ F, then let F∗ = F \ ker F and F• = ker F, so that F∗ ∈ F ∗ , F• ∈ F • and F∗ ∪ F• = F , proving that F ⊃ F ∗ ∧ F •. It is clear that F ∗ ∨ F • = 2X , because ker F ∈ F • and X \ ker F ∈ ∗ F .  ∗

Remark II.4.2. An ultrafilter U is either principal or free. As U = U ∧U • , then by the maximality of U, either U = U ∗ and U • is improper, or U = U • and U ∗ is improper. If U is a principal ultrafilter on X, then there is x ∈ X such that U = x↑ . In fact, if U = {A}↑ and A = A0 t A1 with A0 6= ∅ and A1 6= ∅, then either A0 ∈ U or A1 ∈ U, which contradicts U = {A}↑ . ∗

12If

ker F = ∅ then F • = 2X ; if ker F ∈ F then F ∗ = 2X .

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A

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0, then ]x − ε, x + ε[ ∈ F0 ∩ F1 for each ε > 0. Non-finitely stable convergences will occur rather seldom in our investigations. Most natural examples of convergences are finitely stable. If ξ is a convergence on a set X, then we define x ∈ limMξ F to the effect that x ∈ limξ F and F is an ultrafilter. It is straightforward that Mξ is a convergence, called the ultrafilter modification of ξ. Proposition III.1.6. If ξ is a convergence on a set X, then Mξ is a non-finitely stable convergence if ξ 6= Mξ.

Proof. If there exists a non-maximal filter F and x such that x ∈ limξ F, then there exists an ultrafilter U ⊃ F and U = 6 F. Therefore, there exists U ∈ U \F, that is, F \U 6= ∅ for every F ∈ F, equivalently, F ∨ (X \ U ) is non-degenerate. Therefore there exists an ultrafilter W ⊃ F ∨ (X \ U ), and obviously U 6= W. Then x ∈ limξ U ∩ limξ W = limMξ U ∩ limMξ W, but limMξ (U ∧ W) = ∅, because U ∧ W is not an ultrafilter.  Example III.1.7 (A non-finitely stable convergence). Let θ be a convergence defined on N ∪ {∞} so that ∞ ∈ limθ F if ker F ⊂ {∞} and either F ∗ includes the cofinite filter of the set {n ∈ N : 2 - n} of odd numbers, or the cofinite filter of {n ∈ N : 2 | n} of even numbers. This is a non-finitely stable convergence, because ∞ ∈ limθ (2n)n , and ∞ ∈ limθ (2n + 1)n , but limθ (N)0 = ∅. Definition III.1.8. The vicinity filter Vξ (x) of ξ at x is the infimum of all filters ξ-converging to x, that is, ^ Vξ (x) := ξ − (x). A vicinity filter Vξ (x) need not ξ-converge to x.

Example III.1.9. Recall the standard sequential convergence σR from Example III.1.3. By virtue of Exercise III.7.1, the vicinity filter VσR (x) of σR at x, is generated by {]x − ε, x + ε[: ε > 0},

hence is not sequential by Example II.2.23. Therefore, x ∈ / limσR VσR (x).

Definition III.1.10 (Pretopologies). A convergence ξ is called a pretopology if ξ − (x) has a coarsest filter for each x ∈ |ξ|. Obviously,

Proposition III.1.11. A convergence ξ on X is a pretopology if and only x ∈ limξ Vξ (x) for each x ∈ X.

38

III. CONVERGENCE OF FILTERS

Discrete and chaotic convergences are pretopologies. The standard convergence νR from Example III.1.4 is a pretopology, because the vicinity filter VνR (x) converges to x. The standard sequential convergence σR from Example III.1.3 is not a pretopology, because, by Example III.1.9, its vicinity filters do not converge. Incidentally, observe that (III.1.4)

VσR (x) = VνR (x).

Definition III.1.12 (Isolated point). An element x of X is called ξ-isolated (or, simply, isolated, if ξ is implicit) if x ∈ limξ F implies that F = x↑ . In other words, a convergence is discrete if and only if all the elements (of the underlying set) are isolated. Definition III.1.13 (Free and Hausdorff convergences). A convergence ξ is said to be free (or T1 ) if (III.1.5)

limξ x↑ ⊂ {x};

it is called Hausdorff (or T2 ) if limξ F is at most a singleton for each filter F, that is, (III.1.6)

{x0 , x1 } ⊂ limξ F =⇒ x0 = x1 .

Free convergences are traditionally called T1 . In order to facilitate the reading to those accustomed to the traditional term, I will be recalling the traditional name T1 — along with the current name free. Proposition III.1.14. A convergence ξ is free if and only if x ∈ limξ F implies that ker F ⊂ {x}. Proof. If a convergence ξ on X is such that x0 ∈ limξ F and there is x 6= x0 such that x ∈ ker F, then x↑ ⊃ F and thus x0 ∈ limξ x↑ , that is, ξ is not free. Conversely, if ξ is not free, then there exist x 6= x0 such that x ∈ limξ x↑0 . Thus x0 ∈ ker x↑0 6⊂ {x}.  By the condition (centered), the inclusion in (III.1.5) is, in fact, the equality. Of course, each Hausdorff convergence is free, because (III.1.6) implies (III.1.5) for F = x↑ . It is clear that the discrete convergence ιX is Hausdorff for each set X. On the other hand, the chaotic convergence oX is free (alias T1 ) if and only if card X = 1. By the way, card X = 1 implies that ιX = oX = ξ for each convergence ξ on X. Both convergences of the real line from Examples III.1.3 and III.1.4 are Hausdorff.

III.1. THE CONCEPT OF LIMIT

39

Definition III.1.15. If ξ is a convergence on X, and ∅ 6= V ⊂ X, then the restriction ξ|V of ξ to V is defined by (III.1.7)

lim ξ|V F := V ∩ limξ F,

where F is a filter on V .

As in (III.1.7) F ∈ FV , it follows that it is a filter-base of a filter on X, that is, F ↑ ∈ FX. Nonetheless, limξ F above makes sense, because of (III.1.2). It is straightforward that Proposition III.1.16. Each restriction of a Hausdorff convergence is Hausdorff, and each restriction of a free convergence is free. Example III.1.17 (Cofinite convergence). We define the cofinite convergence of an infinite set X to be a pretopology θ such that X = limθ (X)0 . Then {x} = limθ x for each x ∈ X. By (isotone), x ∈ limθ F if and only if F ⊃ (X)0 ∧ x↑ for each x ∈ X. Hence, θ is free (alias T1 ), but not Hausdorff. ↑

Definition III.1.18. A convergence ξ is called T0 if it distinguishes the points, that is, if x0 6= x1 =⇒ ξ − (x0 ) 6= ξ − (x1 ).

Of course, each free convergence is T0 , because if ξ is free, then ∈ ξ − (x1 ) \ ξ − (x0 ) for any x0 6= x1 . The chaotic convergence on a set of cardinality greater than 1 is an example of a convergence that is not T0 . x↑1

Definition III.1.19 (Density). Let ξ be a convergence on X. A subset D of X is called ξ-dense (or simply, dense) if for each x ∈ X there exists a filter F such that D ∈ F and x ∈ limξ F. The density d(ξ) of convergence ξ is the least cardinality of a ξ-dense set. Traditionally a convergence is called separable if its density is countable. Notice that each convergence on a countable set is separable. A subset A of X is ιX -dense (where ιX is the discrete convergence of Example III.1.1) if and only if A = X. It is oX -dense (where oX is the chaotic convergence of Example III.1.2) if and only if ∅ 6= A.

Example III.1.20 (A dense subset of the real line). The set Q of the rational numbers is νR -dense, where νR is the standard convergence on R of Example III.1.3. Let r ∈ R. For every ε > 0, there is n < ω, such that n1 < 2ε, so that ]r − ε, r + ε[ ∩{ nk : k ∈ Z} = 6 ∅. Therefore, ]r − ε, r + ε[ ∩Q 6= ∅ for each ε > 0, so that r ∈ limνR (VνR (r) ∨ Q).

40

III. CONVERGENCE OF FILTERS

III.2. A pointwise perspective Recall that ξ (x) is the set of all filters ξ-converging to x. As we have already written (III.1.3), −

F ∈ ξ − (x) ⇐⇒ x ∈ limξ F.

It is instructive to look at a convergence at each of its points. Of course, each proposition formulated in terms of limξ can be reformulated in terms of ξ − . Although passing from one form to another description provides no new information, it offers a different, sometimes illuminating, perspective. Let us reformulate the axioms of convergence from the pointwise viewpoint. A relation ξ between filters on X and elements of X is a convergence whenever, for each x ∈ X,

(isotone)

(centered)

G ⊃ F ∈ ξ − (x) =⇒ G ∈ ξ − (x), x↑ ∈ ξ − (x).

A convergence ξ is finitely stable whenever (III.2.1)

F0 , F1 ∈ ξ − (x) =⇒ F0 ∧ F1 ∈ ξ − (x).

A convergence ξ is Hausdorff if and only if, for each x0 , x1 , (Hausdorff)

x0 6= x1 =⇒ ξ − (x0 ) ∩ ξ − (x1 ) = ∅.

Observe that (isotone) and (centered) are pointwise properties, that is, they concern each point x of a convergence space separately (from the other points), while (Hausdorff) is not. In contrast, being free (alias T1 ) is a pointwise property, because ξ is a free convergence on X if and only if ξ − (x) \ {x↑ } consists of free filters for each x ∈ X. Recall that an element x of |ξ| is called isolated (3) if (III.2.2)

ξ − (x) = {x↑ }.

Example III.2.1. Let ζ be a convergence on the real line such that ζ − (x) := νR− (x) if x is rational, and ζ − (x) := σR− (x) if x is irrational. We readily see that ζ is a finitely stable convergence. It is pretopological at rational points (or rational points are pretopological ), and not pretopological at irrational points. Definition III.2.2 (Pavement). A collection of filters D ⊂ ξ − (x) is called a pavement of ξ at x if for each filter F ∈ ξ − (x), there is D ∈ D such that F ⊃ D. It follows from Definition III.1.10 that

3The

term is motivated by the fact that no filter from outside of {x} converges to x.

III.3. ORDER ON CONVERGENCES

41

Proposition III.2.3. A convergence ξ on X is a pretopology if and only if for each x ∈ X, there exists a pavement of ξ at x consisting of one filter. Corollary III.2.4. If a finitely stable convergence has a finite pavement at every point, then it is a pretopology. Example III.2.5 (An uncountable pavement). As the standard sequential convergence σR from Example III.1.3 is finitely stable, but not a pretopology, it has infinite pavements. Actually, each pavement of σR is uncountable at every point. Indeed, suppose that {En : n < ω} ⊂ S is a σR -pavement at 0. Then for every S n < ω, there is a countable set En such that En ∈ En . Hence, E := n νR (that is, σR ≥ νR , but σR 6= νR ), where σR is the standard sequential convergence from Example III.1.3, and νR is the standard convergence from Example III.1.4. Proposition III.3.2. The discrete convergence ιX is the finest, and the chaotic convergence oX is the coarsest convergence on X.

42

III. CONVERGENCE OF FILTERS

Proof. Let ξ be a convergence on X. Indeed, x↑ is the only filter that belongs to ι− (x) for every x ∈ X. By (centered), ι− (x) ⊂ ξ − (x) for every x ∈ X and convergence ξ on X, that is, ι ⊂ ξ. On the other hand, limξ F ⊂ X = limo F for each F ∈ FX, that is ξ ⊂ o.  The order on the set of convergences on X is not linear, unless X is a singleton. Indeed,

Example III.3.3. If x0 , x1 ∈ X, then let limξ0 x↑0 := {x0 , x1 } and limξ0 x↑ := {x} for x 6= x0 , and limξ1 x↑1 := {x0 , x1 } limξ1 x↑ := {x} for x 6= x1 . Then x1 ∈ limξ0 x↑0 \ limξ1 x↑0 and x0 ∈ limξ1 x↑1 \ limξ0 x↑1 , so that ξ0 and ξ1 are incomparable whenever x0 6= x1 .

The convergences in the preceding examples are not free. Let us give an example of two free incomparable convergences.

Example III.3.4. If X is infinite, x0 , x1 ∈ X, then let π0 be a finitely stable convergence on X such that each x 6= x0 is isolated and x0 ∈ limπ0 (X)0 , and π1 be a finitely stable convergence such that each x 6= x1 is isolated and x1 ∈ limπ1 (X)0 . If x0 6= x1 , then x0 ∈ limπ0 (X)0 \ limπ1 (X)0 , and x1 ∈ limπ1 (X)0 \ limπ0 (X)0 . In this case, π0 ∨ π1 = ι and π0 ∧ π1 is the cofinite convergence of Example III.1.17. Proposition III.3.5. Let ζ and ξ be convergences on a given set so that ζ ≥ ξ. If ξ is free (alias T1 ), then ζ is free. If ξ is Hausdorff, then ζ is Hausdorff.

Proof. If a convergence ξ on X is free, then {x} ⊃ limξ x↑ ⊃ limζ x↑ for each x ∈ X, because ζ ≥ ξ, hence ζ is free. If {x0 , x1 } ⊂ limζ F, then {x0 , x1 } ⊂ limξ F, hence x0 = x1 , because ξ is Hausdorff. Thus ζ is Hausdorff.  Another example is provided by Example III.3.6 (Sorgenfrey line). Let ς be a convergence on the real line to the effect that x ∈ limς F whenever [x, x + ε[ ∈ F

for each ε > 0. We easily check that ς is a Hausdorff pretopology of countable character, because the coarsest filter ς-convergent to x is the filter generated by the filter-base {[x, x + 2−n [: n < ω}. We notice that ς > νR , the standard convergence of the real line, as the coarsest filter νR -convergent to x is generated by { ]x−ε, x+ε[ : ε > 0}. This convergence is also called the right half-open interval convergence, because it is defined by the intervals closed on the left and open on the

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0, is called the left half-open interval convergence. Of course, one of the two Sorgenfrey convergences is transformed to the other by the inversion of the order of the real line (5). Sure enough, neither ς+ ≥ ς− , nor ς− ≥ ς+ . In other words, they are incomparable. Both are finer than the standard convergence ν = νR . Actually, ς+ ∧ ς− = ν and ς+ ∨ ς− = ι,

where the extrema ∧, ∨ are considered within the finitely stable convergences (See Exercise III.7.3).

Proposition III.3.7. The supremum and the infimum exist for every non-empty set Ξ of convergences on X, namely \ (III.3.3) limW Ξ F = limξ F, ξ∈Ξ [ (III.3.4) limV Ξ F = limξ F.

ξ∈Ξ

44

III. CONVERGENCE OF FILTERS

T Proof. Clearly, θ∈Ξ limθ F ⊂ limξ F for each ξ ∈ Ξ. It is enough to observeSthat the intersection defines a convergence on X. Similarly, limξ F ⊂ θ∈Ξ limθ F for each ξ ∈ Ξ, and the union defines a convergence on X.  Example III.3.8. Let νx be a convergence on R defined by ( {F : ∀ε>0 ]r − ε, r + ε[∈ F}, r = x, νx (r) = r↑ r 6= x, V W for each x ∈ R. We easily see that x∈R νx = ν, and x∈R νx = ι.

It is obvious that the supremum of a non-empty set of Hausdorff convergences is Hausdorff. One sees easily that the infimum of (two) Hausdorff convergences need not be Hausdorff.

Lemma III.3.9. The supremum and the infimum of each non-empty set of free (alias T1 ) convergences is free. The finest free convergence on a set X is the discrete convergence ιX , and the coarsest free convergence on X is the cofinite convergence of X (6). W Proof. Let Ξ be a set of free convergences on X, then Ξ is free, because if ξ is a free convergence and θ ≥ ξ then x ∈ limθ F ∈ limξ F, hence ker F ⊂ {x}, and thus θ is free. On the other hand, x ∈ limV Ξ F, then limξ {x}↑ ⊂ {x} for every x ∈ X and each ξ ∈ Ξ, thus limV Ξ {x}↑ ⊂ {x} for every x ∈ X. Accordingly, if ξ is a free convergence on X and ∅ 6= limξ F, then F ∗ ⊃ (X)0 . Therefore, x ∈ limθ0 F whenever F ⊃ (X)0 ∧ x↑ , then θ0 is free, and the coarsest one.  Remark III.3.10. If ζ and ξ are preconvergences on a set X, then we write ζDξ whenever (III.3.2) holds for each for each F ∈ FX. It is straightforward that D is an order on the set of preconvergences on X. Of course, if ζ and ξ are convergences on a set X, then ζ D ξ if and only if ζ ≥ ξ. If now ξ0 is a convergence on X0 and ξ1 is a convergence on X1 , then (III.3.5)

ξ0 D ξ1

means that limξ0 F ⊂ limξ1 F for each F ∈ F(X0 ∪ X1 ). Accordingly, (III.3.5) holds if and only if X0 ⊂ X1 and ξ0 ≥ ξ1 |X0 . 6From

Example III.1.17.

III.4. PRIME CONVERGENCES

45

III.4. Prime convergences Prime convergences have a particularly simple structure: they are entirely determined by a collection of filters converging to one point. Definition III.4.1 (Prime convergences). A convergence is called prime if it has at most one non-isolated point, which is called the pole of that convergence. If x∞ is a pole of a prime convergence ξ, then we say that ξ is prime at x∞ . A discrete convergence ιX is prime. A chaotic convergence oX is prime if and only if X is a singleton. Proposition III.4.2. Each prime free (alias T1 ) convergence is Hausdorff. Proof. If ξ is prime and discrete, then it is obviously Hausdorff. If ξ is prime non-discrete, then there is a non-isolated point x∞ in |ξ| . If, by way of contradiction, there existed x 6= x∞ such that ∅ 6= ξ − (x) ∩ ξ − (x∞ ) = {x↑ } ∩ ξ − (x∞ ) , then x↑ ∈ ξ − (x∞ ) , which means that x∞ ∈ limξ x↑ , contrary to (III.1.5).  We denote by π[x, D] the prime convergence (7), for which {D ∧ x↑ : D ∈ D}

is a pavement at the pole x. If, in particular, D = {D}, then we abridge π[x, D]. Hence, a prime convergence ξ is a pretopology if and only if ξ = π[x, Vξ (x)] for some x ∈ |ξ|.

Example III.4.3. Let X be an infinite set and let x∞ ∈ X. We define the prime cofinite convergence π = π[x∞ , (X)0 ] on X by declaring isolated each x 6= x∞ , and setting x∞ ∈ limπ F whenever F ⊃ (X, x∞ )0 ,

where (X, x∞ )0 is the cofinite filter centered at x∞ from Definition II.4.3. In particular, for each free filter F F ⊃ (X)0 ⊃ (X, x∞ )0 ,

hence each free filter on X converges to x∞ . The cofinite filter (X)0 on an infinite countable set X is free and sequential, while (X, x∞ )0 is sequential, the free part of which is (X)0 and the principal part is x↑∞ . If you are accustomed to the language of sequences, it may be more comfortable to see π[x∞ , (X)0 ] as a convergence on X = {x∞ } ∪ {xn : n < ω} such that 7The

{x∞ } = limπ (xn )n ,

underlying set X is implicit.

46

III. CONVERGENCE OF FILTERS

and (xn )n ∧ x↑∞ is the coarsest filter-base convergent to x∞ . The prime cofinite convergence is a Hausdorff pretopology. Each convergence can be decomposed into prime convergences, that is, Proposition III.4.4. Each convergence is an infimum of prime convergences. Proof. If θ is a convergence, then for each x ∈ |θ| , let θx be a prime convergence on |θ| such that θx− (x) := θ− (x) , and θx− (y) = {y ↑ } for each y 6= x. Then y ∈ limVx∈|θ| θx F if and only if there exists x ∈ |θ| such that y ∈ limθx F, equivalently, F ∈ θx− (y) , that is, either y 6= x, hence F = y ↑ , or y = x, hence F ∈ θy− (y) = θ− (y), which means that y ∈ limθ F.  Similarly to cofinite convergences, we consider (8)

Example III.4.5 (Prime cocountable convergence). Let X be an uncountable set and let x∞ ∈ X. A prime cocountable convergence π[x∞ , (X)1 ] is a pretopology, in which the vicinity filter of a nonisolated point x∞ is the cocountable filter of X centered at x∞ . In fact, Vπ (x) = (X, x∞ )1 := {F ⊂ X : x∞ ∈ F ∧ card(X \ F ) ≤ ℵ0 }.

Example III.4.6. Let X be an infinite countable set, x∞Fand Xn an infinite S subset of X for each n < ω such that X = {x∞ } t nk Xn : k < ω} be a filter-base of a filter G. We define a prime convergence γ on X to the effect that x∞ ∈ limγ F whenever F ⊃ G ∧ x↑∞ . Of course, γ is a pretopology, and its vicinity filter is countably based. Proposition III.4.4 exhibits a decomposition of an arbitrary convergence into prime convergences. On the other hand, each prime convergence can be decomposed into simpler prime convergences, the pavements of which consist of at most two filters each, as in Example III.4.7 below. We shall see in Proposition VI.5.3 that finitely stable convergences can be decomposed into prime pretopologies. Example III.4.7 (Sequential fan). Let {xn,k : n, k < ω} be a set of distinct points, and let x∞ ∈ / {xn,k : k, < ω}. The set X = {x∞ } t {xn,k : n, k < ω}, endowed with the finest convergence, in which x∞ ∈ limπ (xn,k )k for each n < ω, it is called a sequential fan. 8More

generally, if card X ≥ κ ≥ ℵ0 , then a co-κ filter consists of subsets F of X such that card(X \ F ) < κ, is proper.

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latexit


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0, there exists nε < ω such that d(xn , x) < ε for every n > nε .

50

III. CONVERGENCE OF FILTERS

◆ I II

1

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samedi 25 octobre 14

Figure III.5.2. There are 16 equivalence classes of pretopologies on a three-point set. A solid arrow inside an oval represents the convergence of a point (that is, the principal ultrafilter) to another point (all points converge also to themselves, because each pretopology is centered). A dotted arrow goes from a finer to the coarser pretopology. The 16 classes are ranged in rows corresponding to the number of solid arrows. We marked the discrete pretopology ι, the chaotic pretopology o, and a Bourdaud pretopology ¥ (discussed in detail in Section VI.7). Definition III.6.1. If d is a metric on X and ε > 0, then the set (III.6.5)

B (x, ε) = Bd (x, ε) := {y ∈ X : d (x, y) < ε}

is called a strict ball of radius ε centered at x. If d is a metric, then x ∈ limtd F if and only if Bd (x, ε) ∈ F for each ε > 0. Indeed, as d is non-negative, (III.6.4) means that for each ε > 0 there exists F ∈ F such that F ⊂ B (x, ε) . In other words, the filter Vtd (x) := {Bd (x, ε) : ε > 0}↑ is the coarsest filter td-convergent to x.

III.6. METRIZABLE CONVERGENCES

51

Definition III.6.2. A convergence ξ on X is called metrizable if there exists a metric d such that ξ = td. Then d is said to be compatible with the convergence ξ. Of course, each metrizable convergence is a pretopology, which is Hausdorff, because of (III.6.2). Let us summarize for future references. Let ξ be a (metrizable) convergence, and d any compatible metric. Then (III.6.6)

x ∈ limξ F ⇐⇒ ∀ Bd (x, ε) ∈ F. ε>0

Example III.6.3. The convergences from Examples III.1.4 and III.1.1 are metrizable, and those from Examples III.1.3 and III.1.2 are (obviously) not. A metric d on R that metrizes the standard convergence of Example III.1.4 is (III.6.7)

d (x0 , x1 ) := |x1 − x0 | .

The discrete convergence on X (of Example III.1.1) can be metrized by a discrete metric  0, if x0 = x1 , iX (x0 , x1 ) := 1, if x0 6= x1 .

Notice that the ball BiX (x, ε) = {x} if ε < 1 and BiX (x, ε) = X if ε ≥ 1. A metrizable convergence can admit several compatible metrics, that is, the metrics that metrize it. Example III.6.4. Two metrics compatible with the discrete convergence on X = { n1 : n ∈ N1 } with d(r, s) = |r − s| and iX (r, s) := 1 if r 6= s and iX (r, r) := 0. Of course, d ≤ iX , but there is no c > 0 such that iX ≤ c · d. Definition III.6.5 (Euclidean convergence). We call Rm = {(x1 , . . . , xm ) : x1 , . . . , xm ∈ R}

a Euclidean space of dimension m. The functions r Xm |xk − yk |2 , d (x, y) := k=1 Xm s(x, y) := |xk − yk | , k=1

t (x, y) := max {|xk − yk | : 1 ≤ k ≤ m}

are metrics. The first of them is called the Euclidean metric.

52

III. CONVERGENCE OF FILTERS t1 d1 s1

Figure III.6.1. Representations of balls of radius 1 with respect to the three metrics in 2-dimensional Euclidean space. It is straightforward that for each x, y, (III.6.8)

t (x, y) ≤ d (x, y) ≤ s(x, y) ≤ m · t (x, y) ,

as one can infer from Figure III.6.1. Consequently, the three metrics above are equivalent, in the sense that tt = td = ts. Now, it is a simple observation that a filter F converges to x if and only if xk ∈ limν pk [F] for each 1 ≤ k ≤ m (where pk (x) := xk , the k-th factor of x). For future references’ sake, let us introduce the concept of norm in a real vector space X, which is a map k·k : X −→ R+ such that for each x, y ∈ X, and λ ∈ R, kxk = 0 =⇒ x = 0, kλ · xk = |λ| kxk ,

kx + yk ≤ kxk + kyk .

Each norm defines a metric by d(x, y) := kx − yk. In particular, in Euclidean space Rm , where x = (x1 , . . . , xm ), there are three basic norms: r Xm kxk2 := (III.6.9) |xk |2 , k=1 Xm (III.6.10) kxk1 := |xk | , k=1

(III.6.11)

kxk∞ := max {|xk | : 1 ≤ k ≤ m} .

They are related to the metrics just defined by d(x, y) = kx − yk2 ,

s(x, y) = kx − yk1 ,

t(x, y) = kx − yk∞ .

III.7. SUPPLEMENT

53

III.7. Supplement

V Exercise III.7.1. Show that the filter VσR (x) := F∈σ− (x) F has a R filter-base consisting of the strict intervals ]x − ε, x + ε[ with ε > 0.

Solution. It follows from Section II.1, that for each sequential filter F such that x ∈ limσR F, there exists ε > 0 such that ]x−ε, x+ε[∈ F. If E ≈ ϕ is a sequential filter such that there is ε > 0, for which ]x−ε, x+ε[∈ / E. Consequently, {n ∈ N : ϕ(n) ∈]x−ε, / x+ε[} is infinite, and thus x ∈ / limσR E. Exercise III.7.2. Describe $0 ∨ $1 and $0 ∧ $1 .

− Solution. As ($0 ∨ $1 )− (x) = $− 0 (x) ∩ $1 (x), the supremum $0 ∨ $1 − − is the discrete convergence. As ($0 ∧ $1 ) (x) = $− 0 (x) ∪ $1 (x), the infimum $0 ∧ $1 is the chaotic convergence.

Exercise III.7.3. Show that ς+ ∧ς− = νR and ς+ ∨ς− = ι, where the extrema ∧, ∨ are considered within the class of finitely stable convergences. Solution. In fact, by definition, x ∈ limς+ ∧ς− F if and only if x ∈ limς+ F or x ∈ limς− F, that is, either [x, x + ε[∈ F for each ε > 0, or ]x−ε, x] ∈ F for each ε > 0. Or else, the filter-bases {[x, x+ε[: ε > 0} and {]x − ε, x] : ε > 0} converge to x in ς+ ∧ ς− , and since ς+ ∧ ς− is supposed to be finitely stable, also {]x − ε, x] ∪ [x, x + ε[: ε > 0} does. Of course, ]x − ε, x] ∪ [x, x + ε[ = ]x − ε, x + ε[, and {]x − ε, x + ε[: ε > 0} is the coarsest filter-base νR -convergent to x. Now, x ∈ limς+ ∨ς− F if and only if x ∈ limς+ F and x ∈ limς− F, that is, [x, x + ε[∈ F and ]x − ε, x] ∈ F for each ε > 0, which amounts to {x} ∈ F, that is, x ∈ limι F. Exercise III.7.4. Show that the set of irrational numbers is νR dense in R.

Solution. For each q ∈ Q and every n ∈ N1 , the interval ]q− n1 , q+ 1 [ contains an irrational number rn . In fact, by Example I.5.1, each n open interval is equipotent with the whole real line, hence uncountable by Proposition I.5.7. On the other hand, Q is countable by Proposition I.5.5. By Definition II.1, (rn )n νR -converges to q. Exercise III.7.5. Prove (III.6.8). The following example shows that the property T0 is nor preserved under restrictions.

54

III. CONVERGENCE OF FILTERS

Example III.7.6. Let υ be a pretopology on X = {0, 1, 2} to the effect that limυ 0↑ = limυ 1↑ := {0, 1}, and limυ 2↑ := {0, 2}. Accordingly, Vυ (0) = {0, 1, 2}↑ , Vυ (1) = {0, 1}↑ , Vυ (2) = 2↑ ,

so that υ is T0 . However, the restriction υ|{0,1} = o{0,1} is not T0 .

Remark III.7.7. The concept of filter as a family of subsets of a given set, is a special case of a more general notion. Let (A, ≤) be a lattice (see Section II.3). Then F ⊂ A is called a lattice-filter provided that, F 6= ∅, if a ∈ F and x ≥ a, then x ∈ F , and if a0 , a1 ∈ F then the infimum a0 ∧ a1 ∈ F . Of course, a filter F on X is a subset of 2X , for A0 , A1 ∈ F the infimum in (2X , ⊂) is A0 ∧ A1 = A0 ∩ A1 ∈ F, hence F is a lattice-filter on (2X , ⊂). Proposition III.7.8. If ξ is a finitely stable convergence on X, then ξ − (x) is a lattice-filter for each x ∈ X. Let X be a non-empty set.

Proposition III.7.9. If θ ⊂ FX × X, then there exists the finest preconvergence I◦ θ that is coarser than θ. Proof. We define x ∈ limI◦ θ F if there exists E ∈ FX such that x ∈ limθ E and E ≤ F. Then I◦ θ is a preconvergence and I◦ θ ≤ θ. If ξ is a preconvergence such that ξ ≤ θ, then ξ = I◦ ξ ≤ I◦ θ, so that I◦ θ is the finest preconvergence coarser than θ.  It is plain that θ0 ≤ θ1 =⇒ I◦ θ0 ≤ I◦ θ1 , I◦ θ ≤ θ, I◦ (I◦ θ) = I◦ θ.

A preconvergence ξ on a non-empty set X is a convergence if and only if ξ ≤ ιX . Indeed, x ∈ limιX F if and only if F = x↑ , hence ξ ≤ ιX whenever (centered) holds. Hence, for any θ ⊂ FX × X, by Proposition III.7.9, Iθ := I◦ θ ∧ ιX is the finest convergence that is coarser than θ.

Lemma III.7.10. If θ ⊂ FX × X, then there exists the finest convergence Iθ that is coarser than θ.

We call I◦ the preconvergence modifier , and I the convergence modifier. Example III.7.11. Let η ⊂ FR × R be defined by x ∈ limη F whenever there exists a sequence (xn )n of elements of R such that F = {{xn : n > k}}↑ and for each ε > 0 there is k < ω such that |xn − x| < ε

III.7. SUPPLEMENT

55

for every n > k. In other words, this a usual convergence of sequences. Then Iη is the standard sequential convergence from Example III.1.3. As η ≤ ιX , we infer that Iη = I◦ η.

Example III.7.12. If F is a set of filters on a set X with the following properties (10): (1) x↑ ∈ F for each x ∈ X, (2) if F ∈ F and F ≤ G, then G ∈ F . The relation χF defined by ( X F ∈ F, limχF F := ∅ F∈ / F,

is a convergence, called a characteristic convergence of F . If for each pair of filters V F0 , F1 ∈ F implies that F0 ∩ F1 ∈ F , then χF is finitely stable; if H ∈ F for every non-empty subset H of F , then χF is a pretopology.

10The

so-called Cauchy structure of E. Lowen-Colebunders [180].

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Y latexit

X


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latexit


0, there exists F ∈ F such that supr∈F |f (x) − f (r)| < ε for each r ∈ F . As x ∈ limνR F if and only if ]x−δ, x+δ[ ∈ F for each δ > 0, we can rephrase the condition above to the effect that for each ε > 0, there exists δ > 0 such that |r − x| < δ implies that |f (r) − f (x)| < ε. Example IV.2.4. Consider a metric d compatible with a metrizable convergence ξ on X. Let us observe that d(·, x) ∈ C (ξ, νR ) for each x ∈ X. Indeed, by (III.6.3) and (III.6.1), d(y, x) − d(z, x) ≤ d(y, z), hence d(z, x) − d(y, x) ≤ d(y, z), by inverting the role of y and z, so that |d(y, x) − d(z, x)| ≤ d(y, z), which implies the continuity of d(·, x) in virtue of Example IV.2.3. If X is a normed vector space, then kxk = d(x, 0), where 0 is the zero vector of X. Therefore, the norm is continuous.

60

IV. CONTINUITY

Lemma IV.2.5. If f ∈ C (ξ, τ ), then for each x ∈ |ξ|,

(IV.2.3)

f [Vξ (x)] ≥ Vτ (f (x)).

Proof. Indeed, if W ∈ Vτ (f (x)), then f − (W ) ∈ F for each F ∈ ξ (x), because f [F] ∈ τ − (f (x)), that is, f − (W ) ∈ Vξ (x), hence W ⊃ f (f − (W )) ∈ f [Vξ (x)].  −

Consequently, if τ is a pretopology, then f ∈ C (ξ, τ ) if and only if (IV.2.3) holds for each x ∈ |ξ|. If ξ and τ are metrizable pretopologies, and dξ and dτ are any respective compatible metrics, then we can rephrase (IV.2.3) as follows: (IV.2.4)

∀ ∃

∀ dξ (w, x) < δ =⇒ dτ (f (w), f (x)) < ε.

ε>0 δ>0 w∈|ξ|

On the other hand, Proposition IV.2.6. If ξ and τ are metrizable convergences, then f ∈ C (ξ, τ ) if and only if (IV.2.5)

x ∈ limξ (xn )n =⇒ f (x) ∈ limτ (f (xn ))n

for each sequence (xn )n of elements of |ξ| .

Proof. Let ξ and τ be metrizable, and dξ and dτ are respective compatible metrics, and let ε and δ be as in (IV.2.4). If x ∈ limξ (xn )n , then there exists nδ < ω such that dξ (xn , x) < δ for each n > nδ . Hence, by (IV.2.4), dτ (f (xn ), f (x)) < ε for each n > nδ , so that, f (x) ∈ limτ (f (xn ))n . Conversely, if (IV.2.4) does not hold, then there is ε > 0 such that for every n < ω, there exists xn such that dξ (xn , x) < 2−n and dτ (f (xn ), f (x)) ≥ ε in contradiction with (IV.2.5).  As one can guess that, in general, sequences are not sufficient to characterize continuity.

Example IV.2.7. Consider the convergences σR and νR from Examples III.1.3 and III.1.4. Of course, iR ∈ C (σR , νR ), but iR ∈ / C (νR , σR ), because 0 ∈ limνR VνR (0) \ limσR VνR (0) and iR [VνR (0)] = VνR (0) . On the other hand, νR and σR coincide for sequences. Definition IV.2.8 (Homeomorphism). Convergences ξ and τ are said to be homeomorphic, in symbols, ξ≈τ

if there exists a bijective map f ∈ C (ξ, τ ) such that f −1 ∈ C (τ, ξ). Then f is called a homeomorphism. Denote H (ξ, τ ) the set of all homeomorphisms between ξ and τ .

IV.3. CONTINUITY AND ORDER

61

Homeomorphic convergences can be seen as identical copies of the same convergence. Therefore, if we identify a convergence, then, in fact, we identify the class of its homeomorphic copies, so that we use to say that a convergence is the required one “up to a homeomorphism”. If card X = card Y , then ιX ≈ ιY (the discrete convergences) and oX ≈ oY (the chaotic convergences). In fact, any bijection h : X −→ Y is a homeomorphism in both cases. The Sierpiński pretopologies of Example III.5.3 are homeomorphic: $0 ≈ $1 . The map h : {0, 1} −→ {0, 1} given by h(x) := 1 − x, is a homeomorphism. Accordingly, by a Sierpiński pretopology $ we intend anyone of the homeomorphic variants. The Sorgenfrey convergences ς+ , ς− from Example III.3.6 are homeomorphic: h : R −→ R given by h(x) := −x, is a homeomorphism. The standard convergence of the real line (Example III.1.4) and its restriction to an open interval are homeomorphic (2). IV.3. Continuity and order Definition IV.3.1 (Initial convergence). Let f : X → Y and let τ be a convergence on Y . The coarsest convergence on X, for which f is continuous is called the initial convergence with respect to (f, τ ), and is denoted by f − τ . Proposition IV.3.2. A unique initial convergence f − τ exists for each f and τ . Proof. Let Ξ := {ξ ∈ IX : f ∈ C (ξ, τ )}. V The set Ξ is nonempty, because S ιX ∈ Ξ. We observe that f ∈ C ( Ξ, τ ). Indeed, let x ∈ limV Ξ F = ξ∈Ξ limξ F. Hence there is ξ ∈ Ξ such that x ∈ limξ F, hence, by definition, f (x) ∈ limτ f [F].  The following instrumental formula holds: (IV.3.1)

limf − τ F = f − (limτ f [F]).

Proof. Indeed, f ∈ C (ξ, τ ) amounts to f (limξ F) ⊂ limτ f [F], hence to limξ F ⊂ f − (limτ f [F]). On the other hand, (IV.3.1) defines a convergence. Therefore, f ∈ C (ξ, τ ) if and only if limξ F ⊂ limf − τ F for each filter F, which means that f − τ is the coarsest convergence ξ such that f ∈ C (ξ, τ ).  2Because

tan ∈ H (νR]−π/2,π/2[ , νR ) is a homeomorphism.

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latexit


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latexit

x
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latexit

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τ . Then i ∈ C (ξ, τ ), but iξ = ξ > τ .

Example IV.3.10. Consider on [0, 1] the convergence restricted from the real line, and let us call it µ. Let h : [0, 1] −→ [0, 1[ be defined by h(x) := x if x 6= 1, and h(1) := 0. We notice that h− (r) = {r} if 0 < r < 1, and h− (0) = {0, 1}. In other terms, the preimages of points under h are singletons with exception of 0, when it is a doubleton. 3By

definition, for f ∈ Y X and a convergence ξ on X, the final convergence f ξ is defined on Y . If f is not surjective, then Y \ f (X) is independent of ξ. Therefore, each y ∈ Y \ f (X) is f ξ-isolated.

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latexit

V
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latexit


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latexit

X
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latexit


0, or {0} ∪ [1 − ε, 1[ ∈ F. If we denote by τ the pretopology such that Vτ (0) := Vµ (0)∧(Vµ (1)∨ [0, 1[), and Vτ (r) := Vµ (r), then τ is the finest pretopology for which h ∈ C (µ, τ ), that is, h is pretopologically quotient with respect to µ and τ . We observe that h is also finitely quotient map (which is a stronger property than that of pretopologically quotient map) with respect to µ and τ , but is not a convergence quotient map, because hµ > τ . If ∅ 6= V ⊂ X, then the map jV : V −→ X , defined by jV (x) := x if x ∈ V , is called the natural injection. The notion of natural injection is a conceptual artifact, like that of identity map. On a subset V of given set X, we define an injection jV into X so that jV (V ) = V . Artificial as it may seem, it is sometimes really useful. ⌧

jV

f f⇠

Figure IV.3.2. Natural injection jV : V −→ X, and an embedding f : ξ ,→ τ .

Definition IV.3.11 (Subconvergence). Let ξ be a convergence on a set X and let V ⊂ X. The initial convergence jV− ξ is called a subconvergence or a subspace of ξ.

4

The subconvergence jV− ξ and the restriction ξ|V are homeomorphic ( ), that is, jV− ξ ≈ ξ|V . Accordingly,

Definition IV.3.12 (Embedding). An injective map is called an embedding if its domain and its range are homeomorphic.

lim ξ|V F = V ∩ limξ F.

definition, for a filter F on V ,

where the image jV [F] in X of a filter F on V coincides with a filter-base F in X.

V

limj − ξ F = jV− (limξ jV [F]) = V ∩ limξ F,

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latexit

pX (0)
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

latexit


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

n Xn ∈ F, which yields a contradiction. We conclude that S0 (ξ ∨ τ ) > S0 ξ ∨ τ . The same convergences can be used to prove that the pretopologizer is not productive.

Example VI.6.12 (S0 ξ ×τ  S0 (ξ ×τ )). Let ξ and τ be the convergences from Example VI.6.11. By Exercise IV.10.13 and by (VI.6.7), (S0 ξ × τ )|4X = S0 ξ ∨ τ  S0 (ξ ∨ τ ) = S0 ((ξ × τ )|4X ) = S0 (ξ × τ )|4X .

As the inequality of two convergences does not hold on a subset 4X, it does not hold on the whole X × X, hence S0 ξ × τ  S0 (ξ × τ ).

VI.7. INITIAL DENSITY OF THE BOURDAUD PRETOPOLOGY

123

As we have observed in VI.5.4, f ξ need not be a pretopology when ξ is a pretopology. By Defintion IV.3.9, a surjective map f ∈ C (ξ, τ ) is called a pretopologically quotient whenever τ ≥ S0 (f ξ).

In other words, a pretopology τ is a pretopological quotient if is the finest pretopology, for which f is continuous from ξ. Example VI.6.13. An evident instance of a continuous map that is not pretopologically quotient is the identity iX on a set X, on which a non-pretopological convergence ξ is defined. Then i ∈ C (ξ, S0 ξ), but is not pretopologically quotient, because ξ > S0 ξ. VI.7. Initial density of the Bourdaud pretopology We will consider here a special, but very important, pretopology on a three point set, which is called the Bourdaud pretopology, and is denoted by U. Example VI.7.1 (Bourdaud pretopology). Let U be a pretopology defined on {0, 1, 2} by convergence of ultrafilters as follows: limU 0↑ = {0, 1}, lim¥ 1↑ = {0, 1, 2} = lim¥ 2↑ .

(5) As for vicinities,

VU (0) = V¥ (1) = {{0, 1, 2}}, V¥ (2) = {{1, 2}, {0, 1, 2}}.

Accordingly, adhU {0} = {0, 1}, adh¥ {1} = {0, 1, 2} = adh¥ {2}, hence adh¥ (adh¥ {0}) \ adh¥ {0} = 6 ∅. 0 ^o 

@

2

1

Lemma VI.7.2. If ξ is a pretopology on X, then f ∈ C (ξ, U) if and only if adhξ f − (0) ⊂ f − ({0, 1}).

Proof. Let us use one of the conditions of Proposition VI.4.1, say (4). The only significant case is when B = {0}, that is, adhξ f − (0) ⊂ f − (adhU {0}) = f − ({0, 1}).  Theorem VI.7.3. Each pretopology is a supremum of initial convergences with respect to the Bourdaud pretopology.

5Therefore,

limU {{0, 1}}↑ = {0, 1} = limU {{0, 2}}↑ , limU {{1, 2}}↑ = {0, 1, 2}, and limU {{0, 1, 2}}↑ = {0, 1}.

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VI. PRETOPOLOGIES

Proof. For every A ⊂ X, define a map fA : X −→ {0, 1, 2} by   0 x ∈ A, (VI.7.1) fA (x) := 1 x ∈ adhξ A \ A,  2 x ∈ / adhξ A.

Notice that fA is continuous. Indeed, as fA− (0) = A and fA− ({0, 1}) = adhξ A, the continuityWfollows from Lemma VI.7.2. Therefore, ξ ≥ (fA )− U, and thus ξ ≥ A⊂X (fA )− ¥. To see that the equality holds, suppose that x ∈ / limξ F, that is, by Proposition VI.1.5, there is A0 ⊂ X such that A0 ∈ F # and x ∈ / − adhξ A0 . We shall show that x ∈ / adhf U A0 , equivalently by (VI.4.1), A0 fA0 (x) ∈ / adhU fA0 (A0 ). Indeed, by definition of fA0 above, fA0 (A0 ) = {0}, fA0 (x) = 2, and adhU {0} = {0, 1}, that is, fA0 (x) ∈ / adhU fA0 (A0 ). As A0 ∈ F # , also fA0 (A0 )#fA0 [F], hence fA0 (x) ∈ / limU fA0 [F] by (VI.1.6). Therefore, x∈ / limf − U F ⊃ limWA⊂X f − ¥ F. A0

A



Let us extend Definition IV.4.7: Definition VI.7.4 (Initial density). A convergence σ is called initially dense in a class X of convergences if each ξW∈ X is σ-initial, that is, there exists a set F ⊂ C (ξ, σ) such that ξ = f ∈F f − σ. A class X is said to be simple if there exists in X an initially dense convergence. If σ is a convergence initially dense in X, then σ ∈ X, because σ = i− σ, where i stands for the identity map. Corollary VI.7.5. The Bourdaud pretopology is initially dense in the class of pretopologies. Corollary VI.7.6. Each pretopology is the initial convergence with respect to a product of Bourdaud pretopologies. Proof. If ξ is a pretopology on X, then for every A ⊂ X, let fA be given by (VI.7.1). Let YA be a copy of {0, Q 1, 2} and UA be the Bourdaud pretopology on YA . Define F : X −→ A⊂X YA by

(pA ◦ F )(x) := fA (x), Q where pA is the projection of A⊂X YA onto YA for each A ⊂ X. Then, by Theorem VI.7.3, Y _ F −( UA ) = fA − UA = ξ. A⊂X

A⊂X

VI.8. SUPPLEMENT

125

 In view of Definition VI.7.4, Corollary VI.7.7. The class of pretopologies is simple. VI.8. Supplement Proposition VI.8.1. If ρ is a metrizable pretopology, then x ∈ adhρ A if and only if there exists a sequential filter E such that x ∈ limρ E and A ∈ E. Proof. If the condition holds, then x ∈ adhρ A, by definition of adherence. Conversely, if x ∈ adhρ A then there is a filter F such that A ∈ F # and x ∈ limρ F. As each metrizable topology is a pretopology, this implies that Vρ (x) ⊂ F hence F # ⊂ Vρ (x)# . Let d be one of the metrics compatible with ρ, and let Bd (x , ε) be the ball defined by d, as in (III.6.5). As 
Bd x , n1 : n ∈ N1
is a filter-base of Vρ (x) , for each n ∈ N1 , there exists xn ∈ Bd x , n1 ∩ A. Consequently, x ∈ limρ (xn )n and A ∈ (xn )n .  Proposition VI.8.2. Each metrizable pretopology has idempotent adherence. Proof. Suppose that π is metrizable (by a metric d) and let x ∈ adhπ (adhπ A) . By Proposition VI.8.1, there exists a sequence (xn )n on adhπ A such that x ∈ limπ (xn )n . Hence, for each n ∈ N1 , there exists a sequence (xn,k )k on A such that xn ∈ limπ (xn,k )k . If p ∈ N1 , then, in particular, there exists np such that d(xnp , x) < p1 . If rp := p1 − d(xn , x), then there exists kp such that d(xnp ,kp , xnp ) < rp . Therefore d(xnp ,kp , x) < d(xnp ,kp , xnp ) + d(xnp , x) < p1 .
Accordingly, x ∈ limπ xnp ,kp p and xnp ,kp ∈ A for each p ∈ N1 , hence x ∈ adhπ A.  Exercise VI.8.3. Show that formulae (VI.6.2) , (VI.6.3), and (VI.6.4) are equivalent (for every variable that occurs): (1) (2) (3)

S0 (f − τ ) ≥ f − (S0 τ ), f (S0 ξ) ≥ S0 (f ξ),

C (ξ, τ ) ⊂ C (S0 ξ, S0 τ ).

126

VI. PRETOPOLOGIES

Solution. Assume (1) and let τ = f ξ, so that f − (S0 (f ξ)) ≤ S0 (f − f ξ) ≤ S0 ξ by (IV.3.6). Applying f to the obtained inequality, we get S0 (f ξ) ≤ f f − (S0 (f ξ)) ≤ f (S0 ξ), that is, (2). If f ∈ C (ξ, τ ), that is, f ξ ≥ τ , then, by (2), f (S0 ξ) ≥ S0 (f ξ) ≥ S0 τ , hence f ∈ C (S0 ξ, S0 τ ), that is, (3). By (3), f ∈ C (f − τ, τ ) ⊂ C (S0 (f − τ ), S0 τ ), hence from the second equivalence in (IV.3.5) we get (1). Proposition VI.8.4. If Ξ is a set of pretopologies on a set X, then _ (VI.8.1) VW Ξ (x) = Vξ (x). ξ∈Ξ W Proof. As Ξ ≥ ξ for each ξ ∈ Ξ, _ VW Ξ (x) ≥ Vξ (x). ξ∈Ξ W As x ∈ limξ Vξ (x) ⊂ limξ ξ∈Ξ Vξ (x) for each ξ ∈ Ξ, hence x ∈ W W W limW Ξ ξ∈Ξ Vξ (x), and since Ξ is a pretopology, VW Ξ (x) ≤ ξ∈Ξ Vξ (x).  Therefore,

Proposition VI.8.5. If Ξ is a set of pretopologies on a set X, then [ [ \ inhW Ξ A = inhσ Aσ . Σ⊂fin Ξ

T

σ∈Σ

Aσ ⊂A σ∈Σ

Proof. Indeed, x ∈ inhW Ξ A if and only if A ∈ VW Ξ (x), and by (VI.8.1), there exists a finite subset ΣTof Ξ such that for each σ ∈ Σ, there exists Aσ ∈ Vσ (x) such that σ∈Σ Aσ ⊂ A. As Aσ ∈ Vσ (x) amounts to x ∈ inhσ Aσ , the equality holds. 

Proposition VI.8.6. If Ξ is a set of pretopologies on a set X, then \ \ [ adhW Ξ A = adhσ Aσ . Σ⊂fin Ξ A⊂

S

σ∈Σ

Aσ σ∈Σ

Any assignment of filters {W(x) : x ∈ X} such that x ∈ W for every W ∈ W(x), determines the vicinities of a unique pretopology. Indeed, Lemma VI.8.7. If W(x) is a filter on X such that x ∈ ker W(x) for each x ∈ X, then there exists a unique pretopology λ on X such that Vλ (x) = W(x) for every x ∈ X.

Proof. Let x ∈ limλ F provided that F ≥ W(x) for each x ∈ X. Then λ is obviously a pretopology on X and x ∈ limλ W(x). AccordV ingly, W(x) ≥ Vλ (x) = F ≥W(x) F ≥ W(x). 

VI.8. SUPPLEMENT

127

Proposition VI.8.8. If ah : 2X −→ 2X is such that for all A, A0 , A1 ⊂ X,

(VI.8.2)

ah ∅ = ∅,

(VI.8.3)

A ⊂ ah A,

(VI.8.4)

ah(A0 ∪ A1 ) = ah A0 ∪ ah A1 ,

then there exists a unique pretopology λ on X such that for every A ⊂ X, adhλ A = ah A. Proof. For each x ∈ X define an isotone family W(x) of subsets of X by setting A ∈ W(x)# if and only if x ∈ ah A. To see that W(x) is a filter, equivalently W(x)# is a filter-grill, let A0 ∪ A1 ∈ W(x)# . By definition, x ∈ ah(A0 ∪ A1 ), hence by (VI.8.4), either x ∈ ah A0 or x ∈ ah A1 , so that W(x)# is a filter-grill. By (VI.8.3), x ∈ ah {x}, hence {x} ∈ W(x)# and thus x ∈ ker W(x). By Lemma VI.8.7, there exists a unique pretopology λ such that Vλ (x) = W(x), and thus adhλ A = ah A in view of Proposition VI.1.4. 

CHAPTER VII

Topological structures A topology is a pretopology, whose adherence is idempotent. Topologies are by far the most known and studied convergences. Here they come out as special pretopologies. As we anticipated, the class of topologies is harder to handle than that of all convergences or that of pretopologies. This is why we tackle topologies from a broader perspective. Nevertheless, in spite of structural deficiencies of the class of topologies, it remains a core of convergence theory. VII.1. Closed sets, closure Let ξ be a convergence on a set X. A subset A of X is called ξ-closed or simply, closed if limξ F ⊂ A for every filter F such that A ∈ F. Denote by Cξ the family of all ξ-closed sets. Proposition VII.1.1. A set A is ξ-closed if and only if adhξ A ⊂ A. Proof. Suppose that A is ξ-closed and x ∈ adhξ A. By Remark VI.1.3, there exists a filter H such that A ∈ H, and x ∈ limξ H ⊂ A, by the assumption. Conversely, if adhξ A ⊂ A and A ∈ F, then limξ F ⊂ adhξ F ⊂ A by the definition of adherence.  Remark VII.1.2. Mind that ξ-closed sets are defined for an arbitrary convergence ξ, and, in general, adhξ A is not ξ-closed. Observe that (VII.1.1)

ζ ≥ ξ =⇒ Cζ ⊃ Cξ .

In particular, for the discrete convergence ιX , and chaotic convergence oX on a set X, CιX = 2X and CoX = {∅, X}. 128

VII.1. CLOSED SETS, CLOSURE

129

Proposition VII.1.3. If ξ is a convergence on X, then (VII.1.2) (VII.1.3) (VII.1.4)

∅, X ∈ Cξ , \ H ⊂ Cξ =⇒ H ∈ Cξ , [ H ⊂fin Cξ =⇒ H ∈ Cξ .

Proof. Clearly, limξ F ⊂ X for each filter F ∈ FX. No (proper) filter F fulfills ∅ ∈ F, hence limξ F ⊂ ∅ isTemptily fulfilled, proving (VII.1.2). To see (VII.1.3), let x ∈ adhξ ( H), where H ⊂ Cξ . As T H ⊂ H for each H ∈ H, Proposition VI.2.4 (b) implies that x ∈ adhξ H for each H ∈ H, and T thus x ∈ H, because each HS∈ H is ξclosed. Consequently, x ∈ H. If H is finite, and x ∈ adhξ ( H), then it follows by simple induction from Proposition VI.2.4 S (d) that there exists H ∈ H for which x ∈ adhξ H, hence x ∈ H ⊂ H, because H is ξ-closed, that is, (VII.1.4) is fulfilled. 

Example VII.1.4. For the standard pretopology ν of the real line, the intervals [r, s] := {x ∈ R : r ≤ x ≤ s} and [r, +∞[:= {x ∈ R : r ≤ x} and ] − ∞, s] := {x ∈ R : x ≤ s} are ν-closed, as well as their finite unions and locally finite unions (see Definition VI.2.5). Proposition VII.1.5. Let ξ be a convergence on a set X. For every subset A of X, there exists the least ξ-closed set that includes A. Proof. Indeed, the family Cξ (A) := {H ∈ Cξ : A ⊂ H} is nonempty, because X ∈ Cξ (A). By (VII.1.3), \ clξ A := {H ∈ Cξ : A ⊂ H}

is ξ-closed, and, of course, A ⊂ clξ A. Hence, clξ A ∈ Cξ (A), and thus, clξ A is the least ξ-closed set including A.  The set clξ A is called the closure of A. Observe that A is ξ-closed if and only if clξ A ⊂ A.

Proposition VII.1.6. The closure operation fulfills (for each A, B, A0 , A1 ) (a)

clξ ∅ = ∅,

(b)

A ⊂ B =⇒ clξ A ⊂ clξ B,

(c) (d) (e)

A ⊂ clξ A,

clξ (A0 ∪ A1 ) = clξ A0 ∪ clξ A1 , clξ (clξ A) = clξ A.

130

VII. TOPOLOGICAL STRUCTURES

Proof. (a-c) follow immediately from definitions. By (VII.1.4), clξ A0 ∪ clξ A1 is ξ-closed, and obviously includes A0 ∪ A1 , thus includes the least ξ-closed set clξ (A0 ∪ A1 ) including A0 ∪ A1 . The converse inclusion follows immediately from (b), hence (d) holds. As clξ A is ξ-closed, it is equal to the least ξ-closed set that includes it, that is clξ (clξ A) = clξ A, in particular, (e) holds.  Corollary VII.1.7. The family Cξ of all ξ-closed sets constitutes a complete lattice with respect to the inclusion, so that if ∅ = 6 H ⊂ Cξ , then ^ \ _ [ (VII.1.5) H= H, H = clξ H. H∈H

H∈H

Proof. By (VII.1.3) the infimum is obviously equal to the intersection. As a union of closed sets is not necessarily closed, the supremum is the least closed set including it, that is, its closure.  For each A ⊂ |ξ| ,

adhξ A ⊂ clξ A.

Indeed, as A ⊂ clξ A, by (b), adhξ A ⊂ adhξ (clξ A) ⊂ clξ A, because clξ A is ξ-closed. Corollary VII.1.8. The adherence adhξ is idempotent if and only if adhξ = clξ . Proof. By Proposition VII.1.6, clξ is idempotent. Conversely, if adhξ is idempotent, then in virtue of Proposition VII.1.1, adhξ (adhξ A) ⊂ adhξ A means that adhξ A is ξ-closed, hence clξ A ⊂ clξ (adhξ A) ⊂ adhξ A.  VII.2. Open sets, interior, neighborhoods Let ξ be a convergence on a set X. A subset O of X is called ξ-open if X \ O is ξ-closed. Let us denote by Oξ the set of all ξ-open sets. It follows from the definition that Proposition VII.2.1. A subset O of X is ξ-open if and only if for each filter F on X, (VII.2.1)

O ∩ limξ F 6= ∅ =⇒ O ∈ F.

Proof. Indeed, if O is ξ-open and O ∈ / F amounts to X \ O ∈ F # , hence limξ F ⊂ X \ O, because X \ O is ξ-closed. 

VII.2. OPEN SETS, INTERIOR, NEIGHBORHOODS

131

It follows immediately from Proposition VII.1.3 that (VII.2.2) (VII.2.3) (VII.2.4)

X, ∅ ∈ Oξ , [ A ⊂ Oξ =⇒ A ∈ Oξ , \ A ⊂fin Oξ =⇒ A ∈ Oξ .

If ζ ≥ ξ then Oζ ⊃ Oξ . When ξ is a convergence on X, it is convenient to denote, for each subset A of X, (VII.2.5)

Oξ (A) := {O ∈ Oξ : A ⊂ O},

and, in particular, Oξ (x) := Oξ ({x}) for each x ∈ X. Hence, \ Oξ (x). Oξ (A) = x∈A

For instance, OιX = CιX = 2 , and OoX = CoX = {∅, X}. X

Example VII.2.2. A set is νR -open if and only if it is a union of strict intervals ]r, s[, where r ≤ s. Indeed, x ∈ limνR F if and only if ]r, s[ ∈ F for each r and s such that r < x < s. Example VII.2.3. For the (upper) Sorgenfrey line ς+ (that is, the right half open interval convergence) from Example III.3.6, a set A is open if and only if for each r ∈ A, there exists s > r so that [r, s[ ⊂ A. Indeed, r ∈ limς+ F if and only if [r, s[ ∈ F for each s such that r < s. It is straightforward that for r < s, the intervals ] − ∞, r[ , [r, s[ , and [s, +∞[ are both ς+ -open and ς+ -closed; the intervals ]r, s[ and ]s, +∞[ are ς+ -open, but not ς+ -closed. Proposition VII.1.1 says that a set A is ξ-closed if and only if adhξ A ⊂ A. As inhξ A = (adhξ Ac )c , we infer that Proposition VII.2.4. A set A is ξ-open if and only if A ⊂ inhξ A. Recall (VI.3.1) x ∈ inhξ A if and only if A ∈ Vξ (x). Therefore, Corollary VII.2.5. A set A is open if and only if it is a vicinity of each of its elements. Remark VII.2.6. Observe that Corollary VII.2.5 is a particular form of Proposition VII.2.1 to the effect that the condition on convergent filters is replaced by that on vicinity filters. The latter are infima of the former, but generally need not converge.

132

VII. TOPOLOGICAL STRUCTURES

For each subset A of |ξ|, there exists the largest ξ-open set included in A. This set is called the interior of A, and is denoted by intξ A. It is clear that, similarly to (VI.3.2), (VII.2.6)

intξ A = (clξ Ac )c .

In general, intξ A ⊂ inhξ A ⊂ A for each A. Moreover, inhξ is idempotent if and only if intξ A = inhξ A for each A. It immediately follows from Proposition VII.1.6 via (VII.2.6) that Proposition VII.2.7. The operation of interior fulfills (for each A, B, A0 , A1 ) (a)

intξ X = X,

(b)

A ⊂ B =⇒ intξ A ⊂ intξ B,

(c)

intξ A ⊂ A,

(d)

intξ A0 ∩ intξ A1 = intξ (A0 ∩ A1 ),

(e)

intξ A = intξ (intξ A).

By Corollary VII.1.7 and (VII.2.6), Corollary VII.2.8. The family Oξ of all ξ-open sets is a complete lattice with respect to inclusion, so that if ∅ 6= A ⊂ Oξ , then ^ \ _ [ A = intξ ( A), A= A. A∈A

A∈A

Definition VII.2.9 (Neighborhood). Let ξ be a convergence on a set X. A subset V of X is called a ξ-neighborhood (or simply, neighborhood ) of an element x if there exists a ξ-open set O such that x∈O ⊂V. The set of all ξ-neighborhoods of x, is denoted by Nξ (x). By Corollary VII.2.5, Nξ (x) ⊂ Vξ (x) for every convergence ξ, and each x ∈ |ξ|. Notice that (VII.2.7)

V ∈ Nξ (x) ⇐⇒ x ∈ intξ V.

Therefore, each property of interior, listed in Proposition VII.2.7, corresponds to an equivalent property in terms of neighborhood. In fact, (VII.2.7) translates the first four of them to the effect that Nξ (x) is a filter, and x belongs to its kernel. The condition (e) of Proposition VII.2.7 is tantamount to (VII.2.8)

V ∈ Nξ (x) =⇒

∃

∀ V ∈ Nξ (w).

W ∈Nξ (x) w∈W

Indeed, V ∈ Nξ (x) amounts to x ∈ intξ V ⊂ intξ (intξ V ), that is, W := intξ V ∈ Nξ (x), and thus V ∈ Nξ (w) for each w ∈ W .

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n Ak ) is a decreasing sequence of T in virtue of non-empty τ -closed sets, thus there exists x∞ ∈ nn clτ Ak , so that there is an infinite sequence (kn )n such that x∞ ∈ clτ Akn for each n < ω, contrary to the local finiteness of A. Conversely, if there exists an infinite countable set A = {an : n < ω} of distinct points without τ -accumulation points, then the family {{an } : n < ω} is locally finite, but not finite.  In virtue of Lemma IX.1.21, Proposition IX.3.11. Every regular free (alias T1 ) topology of countable weight is normal. Proof. Let B be a countable base for the open sets. Let F be a closed set and let O ∈ OX (F ). By regularity, for each x ∈ F , there

178

IX. TOPOLOGICAL CONCEPTS

exists Bx ∈ B such that x ∈ Bx ⊂ clX Bx ⊂ O. Thus X is normal by Lemma IX.1.21.  Example IX.3.12. Each subspace of the standard topology of the real line νR is Lindelöf, because it has countable weight. The standard topology, however, is not countably compact. Indeed, { ]−n, n[ : n < ω} is an open cover of R without any finite subcover. The standard topology is σ-compact, that is, a countable union of compact sets. Coming back to the Sorgenfrey line topology from Example III.7.3, Example IX.3.13 (Sorgenfrey line). Each compact subset of the Sorgenfrey line topology ς = ς+ is countable. Indeed, if K is ς-compact, then for each r ∈ K, there exists xr < r such that ]xr , r[ ∩K = ∅. Otherwise, there would be an increasing sequence (rn )n of elements of K such that limn→∞ |r − rn | = 0. Therefore K ⊂ ] − ∞, r0 [ ∪

[

n x such that ]x, rx [ ⊂ A. As these intervals are disjoint, they are at most countably many. Hence R \ A is countable and thus Lindelöf. Example IX.3.14 (A compact, non-separable topology of countable character). Consider the unit square X × Y , where X = Y = [0, 1], with the topology of lexicographic order. More precisely, (x0, y0 ) < (x1 , y1 ) if either x0 < x1 , or x0 = x1 , and y0 < y1 for every (x0, y0 ), (x1 , y1 ) ∈ I 2 . We note that this ordered set is a complete lattice. Indeed, if ∅ 6= R ⊂ I 2 , then R is a relation, and R− Y 6= ∅. Let x+ := sup{x ∈ X : x ∈ R− Y } be the supremum of the

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0 such that ([0, ε[ ∪ ]1 − ε]) ∩ C = ∅, hence the segment of the circle corresponding to the angles from ] − 2πε, 2πε[ is disjoint from f (C), or there is 0 < x < 1 such that f − (y) = {x}, thus there is ε > 0 such that ]x − ε, x + ε[ ∩C = ∅, hence the segment of the circle corresponding to the angles from ]y − 2πε, y + 2πε[ is disjoint from f (C). Consider a relation R ⊂ X × Y , a topology ξ on X, and a topology τ on Y . A relation is called upper semicontinuous at x0 whenever for each O ∈ Oτ (Rx0 ), there exists V ∈ Nξ (x0 ) such that RV ⊂ O. It is called upper semicontinuous if it is upper semicontinuous at every element of X (9). Actually, Proposition IX.4.8. A relation R is upper semicontinuous if and only if R− C is closed for each closed set C. Proof. Let R be upper semicontinuous and C be τ -closed. If y ∈ / R− C, equivalently Ry ∩ C = ∅, then by the upper semicontinuity assumption, there is V ∈ Oξ (y) such that RV ∩ C = ∅, equivalently, V ∩ R− C = ∅, that is, R− C is ξ-closed. Conversely, let O ∈ Oτ (Ry). Therefore, C := (X \O) is τ -closed and C ∩ Ry = ∅, equivalently, y ∈ / R− C. By assumption, R− C is ξ-closed, hence V := X \ R− C is a ξ-open neighborhood of y and V ∩ R− C = ∅, equivalently RV ∩ C = ∅, that is, RV ⊂ O.  It follows from Proposition IX.4.8 that

Corollary IX.4.9. A map f is closed if and only if the relation f − is upper semicontinuous. 9It

follows that R− Y is ξ-closed. Indeed, if x ∈ / R− Y then ∅ ∈ Oτ (Rx) so that there exists V ∈ Oξ (x) such that RV = ∅, that is V ∩ R− Y = ∅.

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A surjective map f is called perfect if f is closed, and f − (y) is ξ-compact for each y ∈ |τ | (10). It is plain that the map f from Example IX.4.7 is perfect. Proposition IX.4.10. If f : |ξ| −→ |τ | is perfect, and K is τ -compact, then f − (K) is ξ-compact. S Proof. Let P be a family of ξ-open sets such that P ∈P P ⊃ f − (K). As f − (y)Sis ξ-compact for every y ∈ Y , there exists Py ⊂fin P such that Qy := P ∈Py P ⊃ f − (y), and thus y ∈ (f (Qcy ))c and (f (Qcy ))c S is τ -open for each y ∈ Y . Hence y∈K (f (Qcy ))c ⊃ K, and by compactS ness, there exists a finite subset A of K such that y∈A (f (Qcy ))c ⊃ K. Therefore, \ \ ∅= f (Qcy ) ∩ K ⊃ f ( Qcy ) ∩ K, y∈A y∈A S T and thus y∈A Qcy ∩ f − (K) = ∅, equivalently y∈A Qy ⊃ f − (K). It S follows that y∈A Py is a finite ξ-subcover of P of f − (K).  Proposition IX.4.11. If f : |ξ| −→ |τ | is perfect, and A is a ξ-locally finite family, then f [A] is τ -locally finite.

Proof. As A is ξ-locally finite, for each y ∈ Y and each x ∈ f − (y) there exists Ox ∈ Oξ (x) such that {A ∈ A : A#O} is finite. As f − (y) is ξ-compact, there exists a finite subset Fy of f − (y) such that Qy := S − x∈Fy Ox ⊃ f (y) and {A ∈ A : A#Qy } is finite. As f is closed, by Proposition IX.4.8, for every y ∈ Y , there exists Oy ∈ Oτ (y) such that f − (Oy ) ⊂ Qy . As f − (Oy )#A if and only if Oy #f (A), the family f [A] is τ -locally finite.  Remark IX.4.12. Observe that we have not assumed continuity in the two propositions above. Proposition IX.4.13. Let f ∈ C (ξ, τ ) be a perfect map. If ξ is Hausdorff, then τ is Hausdorff. If ξ is Hausdorff and regular, then τ is (Hausdorff ) regular. Proof. If y0 6= y1 , then f − (y0 ) and f − (y1 ) are ξ-compact and disjoint. Hence there exist O0 ∈ Oξ (f − (y0 )) and O1 ∈ Oξ (f − (y1 )) such that O0 ∩O1 = ∅. As f is closed, the sets Y \f (X \O0 ) and Y \f (X \O1 ) are τ -open neighborhoods of y0 and y1 respectively. Let us show that they are disjoint: [Y \ f (X \ O0 )] ∩ [Y \ f (X \ O1 )] = Y \ [f (X \ O0 ) ∪ f (X \ O1 )] = Y \ f [(X \ O0 ) ∪ (X \ O1 )] = Y \ f (X \ (O0 ∩ O1 )) = ∅.

10Traditional

definition comprises continuity.

IX.5. SUPPLEMENT

183

If F is τ -closed, and y ∈ / F , then f − (y) ∩ f − (F ) = ∅. As f − (y) is − ξ-compact and f (F ) is ξ-closed, there exist O0 ∈ Oξ (y) and O1 ∈ Oξ (f − (F )) such that O0 ∩ O1 = ∅. As before, (f (O0c ))c and (f (O1c ))c are τ -open and disjoint neighborhoods of y and F respectively.  IX.5. Supplement Q For any cardinal κ, let $κ ≈ j∈κ $ be the Sierpiński cube of dimension κ (11). Showing that a topology ξ is $-initial in Theorem VII.5.2, we could have taken only the characteristic functions of the members of a base for the ξ-closed sets. As there is a base of cardinality w(ξ), we conclude that Theorem IX.5.1. Each T0 -topology of weight κ is embeddable in a Sierpiński cube of dimension κ. Definition IX.5.2. A convergence is said to be universal for a class of convergences if each convergence from that class is embeddable into that convergence. Therefore, Corollary IX.5.3. The Sierpiński κ-cube is universal for T0 -topologies of weight κ. Proposition IX.5.4. Let ξ be a convergence on X and f ∈ Y X . Then D is f ξ-closed if and only if f − (D) is ξ-closed. Proof. By definition, D is f ξ-closed if and only if D ⊃ adhf ξ D = D ∪ f (adhξ f − (D)) if and only if f (adhξ f − (D)) ⊂ D. By (IV.1.2) adhξ f − (D) ⊂ f − (D), that is, f − (D) is ξ-closed. Conversely, by applying f and (IV.1.1), f (f − (D)) ⊂ D, we conclude.  Corollary IX.5.5. If ξ and τ are convergences, τ is Hausdorff, and f ∈ C (ξ, τ ), then gr(f ) is (ξ × τ )-closed. Proof. By Remark IV.10.12, gr(f ) = (f × iY )− 4(Y ), hence by Proposition IX.1.2, 4(Y ) is (τ × τ )-closed, and since f × iY ∈ C (ξ × τ, τ × τ ), the preimage (f × iY )− 4(Y ) is (ξ × τ )-closed.  Exercise IX.5.6. Show that if τ is a regular topology, R is upper semicontinuous at x0 and Rx0 is τ -closed, then R is graph-closed at x0 . 11The

Sierpiński cube is called the Alexandrov cube by R. Engelking [97].

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Solution. If x0 ∈ limξ F and y ∈ / Rx0 , then there is a τ -closed neighborhood V of y such that V ∩ Rx0 6= ∅. By upper semicontinuity, there is F ∈ F such that RF ∩ V = ∅. If U ∈ βR[F] is such that y ∈ limτ U, then V ∈ U and thus limτ U ⊂ V , but since R[F] ≤ U, also Y \ V ∈ U, which yields a contradiction. Exercise IX.5.7. Using closed and open sets, show that every compact subset of a Hausdorff topological space is closed.

Solution. Let ξ be a Hausdorff topology on X. If C is ξ-compact and x0 ∈ / C, then for every x ∈ C there exist Ox ∈ Oξ (x) and Px ∈ Oξ (x0 ) such that Ox ∩PS x = ∅. As C is ξ-compact, there T is a finite subset F of C such that C ⊂ x∈F Ox . On the other hand, x∈F Px ∈ Oξ (x0 ) and \ \ [ Px = ∅, Ox ∩ Px ⊂ C∩ x∈F

proving that C is ξ-closed.

x∈F

x∈F

Exercise IX.5.8. Each closed subspace of a normal topology is normal. Solution. Let A be a closed subspace of a normal topology on X. If F0 , F1 are disjoint closed subsets of A, hence of X, then there exist disjoint X-open sets O0 and O1 such that F0 ⊂ O0 and F1 ⊂ O1 . By Proposition VII.3.12, O0 ∩ A and O1 ∩ A are A-open.

Exercise IX.5.9. A topological space ξ is normal if and only if for each ξ-closed set A and U ∈ Oξ (A) there exists W ∈ Oξ (A) such that (IX.5.1)

clξ W ⊂ U.

Solution. As A is ξ-closed and U ∈ Oξ (A), the sets A and X \ U are ξ-closed and disjoint. By normality, there exist W ∈ Oξ (A) and V ∈ Oξ (X \ U ) such that W ∩ V = ∅. Therefore, clξ W ∩ V = ∅, hence X \ clξ W ⊃ V ⊃ X \ U , so that clξ W ⊂ U . Conversely, if A, B are disjoint ξ-closed sets, then A ⊂ X \ B. Hence there is W ∈ Oξ (A) such that clξ W ⊂ X \ B, which means that X \ clξ W ∈ Oξ (B) and W ∩ (X \ clξ W ) = ∅.

Exercise IX.5.10. A topology is compact if and only if each family of closed sets with finite intersection property has non-empty intersection. Solution. Let τ be a topology on X. By definition, τ is compact if S for each P ⊂ Oτ such that P = X, there exists a finite subfamily P ∈P S P0 of P such that P ∈P0 P = X. If we set D := PT c , then we get an equivalent condition: if for each D ⊂TCτ such that D∈D0 D 6= ∅ for every finite subfamily D0 of D, then D∈D D 6= ∅.

IX.5. SUPPLEMENT

185

Exercise IX.5.11 (Cantor condition). A topology is countably compact if and only if \ Cn 6= ∅ n 0. In particular, there exists U ∈ NX (x) such that f (u) < f (x) + ε ≤ F (x) + ε for each u ∈ U . Hence F (u) = sup F (u) ≤ F (x) + ε, that is, F is upper semicontinuous, hence continuous.  Actually, if we adjoin the supremum and the infimum to an equicontinuous family, then the resulting family is also equicontinuous. In the sequel we will use an equivalent fact only for suprema.

X.3. MORE ON METRICS AND METRIZABLE TOPOLOGIES

193

Lemma X.2.7. If F is equicontinuous and sup{f (x) : f ∈ F } < + ∞ for each x ∈ X, then the family is equicontinuous.

{supf ∈T f : ∅ 6= T ⊂ F }

Proof. Let us abridge fT := supf ∈T f for each (not necessarily finite) subfamily T of F . By assumption, for each x ∈ X and every ε > 0, there exists U ∈ NX (x) such that |f (u) − f (x)| < 2ε for each u ∈ U and f ∈ T . If T ⊂ F , then there is f ∈ T such that fT (x)− 2ε < f (x), thus fT (x) − ε < f (x) − 2ε < f (u) ≤ fT (u) for each u ∈ U . On the other hand, if f ∈ T then fT (x) + ε ≥ f (x) + ε > f (u) for each u ∈ U , hence fT (x) + ε ≥ fT (u) for each u ∈ U .  X.3. More on metrics and metrizable topologies Recall that a metric on a non-empty set X is a function d : X × X −→ R+ such that (III.6.1), (III.6.2), and (III.6.3) for each x, y, and z, d(x, y) = d(y, x), d(x, y) = 0 ⇐⇒ x = y,

d(x, z) ≤ d(x, y) + d(y, z).

Recollect as well that a topology ξ is called metrizable if there exists a metric such that ξ = td, that is, (III.6.4) x ∈ limξ F provided that inf F ∈F supy∈F d(x, y) = 0. Accordingly, a metric determines a topology altogether. Although a metric is a single function, but being a function of two variables, it actually operates as a family of functions: if d is a metric on X, then {d(·, x) : x ∈ X} is the family of functions that specifies the topology. More precisely, if ξ = td, then, for every x ∈ X, the ξ-vicinity filter of x is given by (X.3.1)

Vξ (x) = {{w ∈ X : d(w, x) < ε} : ε > 0}↑ .

Metrics are called equivalent if they define the same topology. Note that if d is a metric, then min(d, 1) is an equivalent metric. Therefore, each metrizable topology admits a bounded metric. Lemma X.3.1. If d : X × X −→ R+ is a metric, then for each y ∈ X, (X.3.2)

|d(x0 , y) − d(x1 , y)| ≤ d(x0 , x1 ).

194

X. FUNCTIONAL STUDY OF TOPOLOGIES

Proof. By (III.6.3), d(x0 , y) ≤ d(x0 , x1 )+d(x1 , y), hence d(x0 , y)− d(x1 , y) ≤ d(x0 , x1 ). By interchanging x0 with x1 , we conclude. 

Consequently, each metric is continuous. More precisely, for each y ∈ X, the function x 7→ d(x, y) is continuous. It follows that every strict ball B(x, ε) := {y ∈ X : d(x, y) < ε} is open. Therefore, each vicinity filter of a metrizable convergence consists of open sets. It results that each metrizable convergence is a topology. If d is a ξ-compatible metric, then _ ξ= d(·, x)− ν, x∈|ξ|

that is,

Corollary X.3.2. If d is a compatible metric of a metrizable topology ξ, then ξ is initial with respect to {d(·, ˙x) : x ∈ |ξ|} and the standard topology ν of the real line. Corollary X.3.3. Each metrizable convergence is a Hausdorff topology of countable character. In particular, if ξ is metrizable, then adhξ A = clξ A, because ξ is a topology. Lemma X.3.4. If d is a metric, then (X.3.3)

|d(x0 , y0 ) − d(x1 , y1 )| ≤ d(x0 , x1 ) + d(y0 , y1 ).

Proof. Applying (III.6.3) twice, we get hence

d(x0 , y0 ) ≤ d(x0 , x1 ) + d(x1 , y1 ) + d(y1 , y0 ), d(x0 , y0 ) − d(x1 , y1 ) ≤ d(x0 , x1 ) + d(y1 , y0 ),

and, by inverting the role of x0 and x1 above, where it is useful, we get (X.3.3).  Corollary X.3.5. If d is a ξ-compatible metric, then d ∈ C (ξ × ξ, ν). For a metric d on a set X, and for a non-empty subset A of X, we define a distance function from A, (X.3.4)

d(x, A) := inf{d(x, a) : a ∈ A}.

The triangular inequality (III.6.3) generalizes to (X.3.5)

|d(x0 , A) − d(x1 , A)| ≤ d(x0 , x1 ).

X.3. MORE ON METRICS AND METRIZABLE TOPOLOGIES

195

Proof. By definition (X.3.4), and by (III.6.3), d(x0 , A) ≤ d(x0 , a) ≤ d(x0 , x1 ) + d(x1 , a) for each a ∈ A. Therefore, d(x0 , A) − d(x1 , a) ≤ d(x0 , x1 ) for each a ∈ A. Hence, d(x0 , A) − d(x1 , A) = d(x0 , A) − inf a∈A d(x1 , a) ≤ d(x0 , x1 ).

On inverting the role of x0 and x1 , we conclude that (X.3.5) holds.  Corollary X.3.6. The distance function d(·, A) is continuous, and if ξ = td, then {x : d(x, A) = 0} = clξ A.

Proof. As d(·, A) is continuous, {x : d(x, A) = 0} is ξ-closed, and includes A. Therefore, clξ A ⊂ {x : d(x, A) = 0}. On the other hand, if x ∈ / clξ A, then there exists ε > 0 such that B(x, ε) ∩ A = ∅, equivalently d(x, A) > ε, so that x ∈ / clξ A.  The distance between two subsets A0 and A1 of a metric space (X, d) is defined by dist(A1 , A0 ) := inf d(x, A0 ) = inf{(x0 , x1 ) : x0 ∈ A0 , x1 ∈ A1 }. x∈A1

Proposition X.3.7. If K is a compact subset, and F is a closed subset of a metric space (X, d) and K ∩ F = ∅, then distd (K, F ) > 0.

Proof. By Corollary X.3.6, the function d(·, F ) is continuous, hence there exists x0 ∈ K such that distd (K, F ) = d(x0 , F ) > 0, by Corollary X.3.6, because x0 ∈ / F. 

Proposition X.3.8. Each countable product of metrizable topologies is metrizable.

Proof. Let τn be a topology on a set Xn metrized by a metric dn . As max(dn , 1) is topologically equivalent to dn , we assume without loss of generality that |dn | ≤ 1 for each n < ω . Then X∞ 1 (X.3.6) d(f, g) := dn (f (n), g(n)) n=0 2n Q Q for every f, g ∈ n 0, there exists a finite subset of X such that X ⊂ Bd (F, ε). Proposition X.3.12. Each compatible metric of a countably compact metrizable topology is totally bounded.

Proof. By way of contradiction suppose that there exists ε > 0 such that X \ Bd (F, ε) 6= ∅ for each F ⊂fin X. Hence, by induction, there exists a sequence (xn )n of elements of X such that d(xn , xk ) ≥ ε for each n 6= k. Then {xn : n < ω} is a closed discrete subspace hence is not compact by Proposition IX.3.9.  Corollary X.3.13. Every countably compact metrizable topology is separable. Proof. Let d be a compact metric on X. Then, by (1), for each n < ω, there is a S finite set Fn such that X ⊂ Bd (Fn , 2−n ). Therefore, the countable set n 0, there exists F ∈ F such that the diameter diam F < ε, that is, inf F ∈F diam F = 0. Lemma X.4.1. If F is a d-fundamental filter such that x ∈ adhtd F, then x ∈ limtd F.

Proof. Let x∞ ∈ adhtd F. If ε > 0, then for each F ∈ F, there is xF ∈ F such that d(xF , x∞ ) < 2ε . As F is d-fundamental, there exists F ∈ F such that d(x, xF ) < 2ε for each x ∈ F . Therefore, if x ∈ F , then d(x, x∞ ) ≤ d(x, xF ) + d(xF , x∞ ) < ε, that is, Bd (x∞ , ε) ∈ F, which proves that x∞ ∈ limtd F. 

In particular, a sequence (xn )n is called Cauchy if the corresponding sequential filter is fundamental. Lemma X.4.1 specializes to sequences, as below. As adherence and sequential adherence coincide in metrizable spaces, this specialization is a simple corollary. Nevertheless, we provide a proof for the reader’s convenience.

Corollary X.4.2. If a Cauchy sequence has a convergent subsequence, then it is convergent. Proof. Let (xn )n be a Cauchy sequence, and let (xnk )k converge to x. If ε > 0, then there exists nε < ω such that d(xn , xm ) < 2ε for each n, m > nε , and there exists kε < ω such that d(xnk , x) < 2ε for k > kε . Consequently, for n > nε and m = nk > nε with k > kε , d(xn , x) ≤ d(xn , xnk ) + d(xnk , x) < ε.



A metric is said to be complete if each fundamental filter is adherent (4). Proposition X.4.3. The following statements are equivalent for a metric space: 4Recall

that a filter F is ξ-adherent if adhξ F 6= ∅.

X.4. COMPLETELY METRIZABLE TOPOLOGIES

(1) (2) (3) (4)

199

a metric is complete, each fundamental filter converges, each Cauchy sequence converges, each Cauchy sequence is adherent.

Proof. (1) ⇒ (2). If d is a complete metric on X, then, by definition, adhtd F 6= ∅ for each d-fundamental. By Lemma X.4.1, limtd F 6= ∅. (2) ⇒ (3). In particular, this holds for every sequential filter. (3) ⇒ (4). Every convergent filter is adherent. (4) ⇒ (1). Let F be d-fundamental. Then for each n < ω, there is Fn ∈ F such that diam(Fn ) < 2−n . We can assume that the sequence (Fn )n is decreasing, for otherwise, we replace Fn+1 by Fn+1 ∩ Fn . If xn ∈ Fn , then (xn )n is a Cauchy sequence such that (xn )n ≥ (Fn )n , hence by assumption ∅ 6= adh(xn )n ⊂ adh(Fn )n . As (Fn )n is a filter-base of a fundamental filter ∅ 6= lim(Fn )n by Lemma X.4.1. As (Fn )n ≤ F, also ∅ 6= lim F ⊂ adh F.  It is straightforward that Proposition X.4.4. Each closed subset of a complete metric space is complete. Here is a classical characterization of completeness. Theorem X.4.5 (Cantor). A metric space is complete if and only if every decreasing sequence of non-empty closed sets with the diameter tending to 0 has non-empty intersection. Proof. Assume completeness, and let (Fn )n be a decreasing sequence of non-empty closed sets with limn→∞ diam Fn = 0. If xn ∈ Fn , then for ε > 0 there is n < ω such that diam Fn < ε. As xk ∈ Fn for each k ≥ n, the sequence (xn )n is Cauchy, hence convergent. Consequently T lim(xk )k ⊂ Fn , because Fn is closed for each n < ω. Therefore ∅ 6= n m} is closed for each m < ω. Indeed, if x ∈ cl Fm0 , then there is a strictly increasing subsequence (nk )k such that x ∈ lim(x T nk )k , which is contradictory by Lemma X.4.1. On the other hand, m m} : m < ω} is a filter-base of the cofinite filter of X. As diam{2−n : n > m} = 2−m−1 , the filter is dX -fundamental, but it is not convergent. Let iX (2−n , 2−k ) = 1 if n 6= k, and iX (2−n , 2−n ) = 0 for each n, k < ω. This is also a compatible metric of the discrete topology on X. Therefore, the only iX -fundamental filters are principal ultrafilters, which are convergent. Therefore, the topology is completely metrizable. Therefore, the identity map i ∈ C (νX , νX ) maps a complete metric space onto a non-complete metric space. It follows that the continuous image of a complete metric space need not be complete; for the same reason, Cauchy sequences are not preserved under continuous maps. On the other hand, Proposition X.4.9. Each compatible metric of a metrizable compact topology is complete. Proof. If d is such a metric, and (xn )n is a Cauchy sequence, then by compactness, there exists a convergent subsequence (xnk )k . By Corollary X.4.2, (xn )n is convergent.  Proposition X.4.10. The product metric (X.3.6) X∞ 1 d(f, g) := dn (f (n), g(n)) n=0 2n Q on n∈N Xn is complete if and only if dn is complete for each n < ω.

Proof. Let d be complete, and let (xk )k be a Cauchy sequence in Q Xn . For an arbitrary ψ ∈ m∈N Xm , define a sequence (ϕk )k by ( xk m = n, ϕk (m) := ψ(m) m 6= n.

X.4. COMPLETELY METRIZABLE TOPOLOGIES

201

As d(ϕk , ϕl ) = 2−n dn (xk , xl ), Q the sequence (ϕk )k is Cauchy, hence convergent to an element ϕ of m∈N Xm . Therefore, (xk )k converges to ϕ(n). Conversely, let dn be complete for each n < ω, and let (ϕk )k be a Cauchy sequence in the product, hence by (X.3.6), (ϕk (n))k is dn -Cauchy, hence convergent to some xn ∈ Xn for every n < ω. Consequently, (ϕk )k converges to ϕ, given by ϕ(n) := xn .  Corollary X.4.11. A countable product of metrizable topologies is completely metrizable if and only if every factor is completely metrizable. In particular, the Baire space of Example X.3.9 is completely metrizable. By Proposition X.4.4, each closed subset of a completely metrizable space is completely metrizable. Each open subset of a completely metrizable space is completely metrizable. Indeed, by Lemma X.10.4 to come, each open subset of a topology ξ is homeomorphic to a closed subset of ξ × νR , which is completely metrizable by Corollary X.4.11 and Lemma X.4.7. Definition X.4.12. A set is called a Gℵ0 -set (traditionally, Gδ -set) if it is a countable intersection of open sets. A set is called a Fℵ0 -set (traditionally, Fσ -set) if it is a countable union of closed sets. Proposition X.4.13. Each Gℵ0 -subset of a completely metrizable topology is completely metrizable. Proof. By Lemma X.10.5 to come, each Q Gℵ0 -subset of a topology ξ is homeomorphic to a closed subset of ξ × n 0 there is nε < ω such that kfn − fk k < ε for each n, k > nε . In particular, (X.4.1)

|fn (x) − fk (x)| < ε

for each n, k > nε and for each x ∈ X. Therefore, the sequence (fn (x))n is Cauchy. By Lemma X.4.7, there exists a subsequence (fnp (x))p convergent to an element f (x) of R. By replacing in (X.4.1) k by np , we conclude that |fn (x) − f (x)| ≤ ε for every x and each n > nε .

202

X. FUNCTIONAL STUDY OF TOPOLOGIES

Consequently, kfn − f k = supx∈X |fn (x) − f (x)| ≤ ε

and each n > nε . Therefore, (fn )n uniformly converges to f .



The following theorem is one of the most instrumental results in topology and analysis. The term category theorem refers to the sets of first category that are countable unions of nowhere dense sets (Definition IX.5.18 to come). In these terms, the following theorem says that completely metrizable spaces are not of first category. Theorem X.4.15 (Baire category theorem). The intersection of a sequence of open dense subsets of a completely metrizable space is dense. Proof. Fix a complete metric compatible on X. For each n < ω, let Qn be an open dense subset of X. Let x ∈ X and let r > 0. By density, there exists x1 ∈ Q0 ∩ B(x, r), and since Q0 is open, there is 0 < r1 < 2r such that cl B(x1 , r1 ) ⊂ Q0 ∩ B(x, r). Suppose that we have found sequences r0 = r, r1 , . . . , rn and x0 := x, x1 , . . . , xn such that 0 < rk+1 < r2k and (X.4.2)

cl B(xk+1 , rk+1 ) ⊂ Qk ∩ B(xk , rk )

for k < n < ω. Then there exist xn+1 and rn+1 such that (X.4.2) for k = n. As (rn )n tends to 0, by Theorem X.4.5 of Cantor, \ \ Qn , cl B(xn , rn ) ⊂ ∅ 6= 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

latexit


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

0 such that for each x ∈ [−ε, ε], there exists δx > 0 such that {x} × [−δx , δx ] ⊂ V , while W ∈ Vξv (0, 0) whenever there is δ > 0 such that for each y ∈ [−δ, δ], there exists εy > 0 such that [−εy , εy ] ⊂ W . Of course, ξ 2 ≤ ξh ∧ξv ≤ ξ, hence Tξ = Tξh = Tξv .

Let ξ be a pretopology. We denote ξ 0 := ι, and ξ n+1 := ξξ n for each n < ω. Accordingly, ξ n ≥ ξ n+1 for each n < ω. In general, an infimum of pretopologies is not a pretopology, so that ^ ^ {ξ n : n < ω} = S0 {ξ n : n < ω}, S0

that is,

7See

[ adhVn 0, the filter eλ [F, f∞ ] = {{eλ (f, f∞ ) : f ∈ F } : F ∈ F} on R+ tends to 0. Of course, eλ ([f ]µ , [g]µ ) := eλ (f, g) is well-defined. Consequently,

552

XXII. IMPLEMENTATIONS AND REFINEMENTS

Proposition XXII.9.18. Convergence in measure is the finest convergence, for which the functionals {eλ : λ > 0} are continuous.

In other words, mµ is the initial convergence with respect to a family of real-valued functions {eλ (·, f ) : λ > 0, f ∈ M}, and thus [mµ ]µ is a Hausdorff, functionally regular (alias completely regular) topology of countable character (see, e.g., [83]). If µ is the Lebesgue measure on a bounded closed interval I, then the topology mµ is separable. Indeed, if µ{|h| > λ} < ε, then there is a simple function s valued in Q and with the intervals with rational ends such that µ{|s| > λ} < ε. There are countably many of them. As we can also take λ, ε ∈ Q, we conclude that the topology mµ is separable. It follows from the Urysohn metrization theorem (e. g., [83, p. 204]) that mµ is metrizable. Corollary XXII.9.19. If µ is the Lebesgue measure on a bounded closed interval, then convergence in measure [mµ ]µ (on equivalence classes) is a metrizable topology. Recall the Urysohn modification of ξ, denoted by Uξ (XVI.3.6): for each sequential filter E, \ [ limξ H. (XXII.9.6) limUξ E := G∈SE

H∈SG

If E is a sequential filter and x ∈ limξ E, then x ∈ limξ H for every filter H ⊃ E, hence limξ E ⊂ limUξ E for each E and ξ, so that ξ ≥ Uξ for each sequentially founded convergence ξ. Accordingly, ξ is a Urysohn convergence whenever Uξ ≥ ξ. By Lemma XXII.9.7, Corollary XXII.9.20. Sequentially founded convergence in measure is the Urysohn modification of the sequentially founded convergence almost everywhere. Proof. Indeed, if a sequential filter E converges to f in measure, then there exists F ∈ SE that converges to f a.e. Hence each G ∈ SF also converges to f a.e., that is, if E converges to f in measure, then it also does in the Urysohn modification of the convergence a.e. On the other hand, convergence a.e. entails convergence in measure.  It follows from Corollary XXI.5.7 that each sequentially founded convergence almost everywhere is a pseudotopology. Hence, (XXII.9.7) However,

Seqeµ = S(Seqeµ ) > U(Seqeµ ) = Seqmµ .

XXII.9. MEASURE-THEORETIC CONVERGENCES

553

Proposition XXII.9.21. Let µ be the Lebesgue measure on [0, 1]. The countably carried modification of almost uniform convergence aµ is not a pseudotopology, that is, Eeµ > S(Eeµ ). Proof. Consider the closed unit interval with Lebesgue measure. Following [126, p. 94], let fk,n ∈ [0, 1][0,1] be the characteristic function of [ n−1 , nk ] for 0 < n ≤ k, 0 6= k ∈ N. Let F be a family of subsets k of {0, 1}[0,1] defined by F ∈ F whenever {(k, n) : fk,n ∈ / F } is finite (20). Then F is a sequential filter that does not converge to 0 almost everywhere, equivalently almost uniformly, because {(k, n) : fk,n (x) = 1} is infinite for each x ∈ [0, 1] (21). I claim that each ultrafilter U finer than F converges to 0 almost uniformly. Let π : {0, 1}[0,1] −→ [0, 1] be any selection π(χA ) ∈ A. Then π(U ) := {π(f ) : f ∈ U }, and π[U] := {π(U ) : U ∈ U}} is an ultrafilter-base on [0, 1]. By compactness of [0, 1], there exists xU such that π[U] converges to xU . Hence, for each ε > 0, there is U ∈ U such that π(U ) ⊂ (xU − 2ε , xU + 2ε ). On the other hand, there is F ∈ F such that if fn,k ∈ F , then k1 < 2ε , nk ]. Consequently, [ n−1 , nk ] ⊂]xU − ε, xU + ε[ for each and π(fn,k ) ∈ [ n−1 k k (k, n) such that fk,n ∈ U ∩ F . Therefore U converges uniformly to 0 out of ]xU −ε, xU +ε[ for each ε > 0, hence converges to 0 almost uniformly. Therefore, F converges to 0 in the pseudotopological modification of the countably carried modification of almost uniform convergence.  Remark XXII.9.22. In the proof, we construct a filter F such that 0 ∈ limSeµ F \ limeµ F. As F is sequential, limeµ F = limSeqeµ F. However, it follows from the construction that limSeµ F = 6 limSSeqeµ F. Of course, the considered ultrafilters are countably carried, because they include the sequential filter F, but none includes a sequential filter convergent to 0. Remark XXII.9.23. Notice that a filter is countably carried if and only if every finer ultrafilter is countably carried. Therefore, SE = ES. Consequently, Eaµ > S(Eaµ ) implies aµ > Saµ . Hence, by Proposition XXII.9.17 and Corollary XXII.9.19, if µ is the Lebesgue measure on [0, 1], then aµ > Saµ ≥ mµ = Tmµ . By Propositions XXII.9.2 and XXII.9.4, then Seq aµ is a common restriction to sequentially founded convergences of convergences almost uniform and almost everywhere, and thus is measure-theoretic. 20In

other words, F is the cofinite filter on the set X := {fn,k : 0 < n ≤ k, 0 6= k ∈ N}, or else corresponds to any order of type ω on X. 21In fact, it converges nowhere.

554

XXII. IMPLEMENTATIONS AND REFINEMENTS

This can be also see directly, because each Seq aµ -convergent filter F contains a countable set, say F0 . Therefore, if there is S ∈ S such that µ(Ω \ S) = 0 so that 0 ∈ lim F(x) for each x ∈ S, and ϕ(h) = h µ-a.e. S for each h ∈ M, then 0 ∈ lim ϕ[F](x) for each x ∈ S0 := S ∪ f ∈F0 {|ϕ(f ) − f |}, and µ(Ω \ 0) = 0. The following example highlights the difference between Seqeµ and Eeµ . It shows that while the former is a pseudotopology finer than Seq mµ (XXII.9.7), the latter is not, and S(Eeµ ) is not finer than E mµ . In particular, the countably carried modification of convergence almost everywhere is not sequentially founded. Example XXII.9.24 (22). Consider the Lebesgue measure on [0, 1]. S For each n < ω, let Kn consist of the sets of the form i∈I [i2−n , (i + 1)2−n ], where I ⊂ {0, . . . , 2n − 1} such that S card I = n. Accordingly, µ(K) = 2nn for each K ∈ Kn . Let X := n 0, and P := {Pε : ε > 0}. A filter 1For

example, R. Engelking, [97, Theorem 3.9.2]. 556

XXIII.1. FUNDAMENTAL FILTERS AND COMPLETENESS

557

F is P -fundamental if and only if for each ε > 0, there is x ∈ X such that Bd (x, ε) ∈ F, that is, if and only if F is a d-Cauchy filter: (XXIII.1.1)

inf F ∈F diam F = 0.

Definition XXIII.1.3 (Completeness). A convergence ξ is said to be P -complete whenever each P -fundamental filter is ξ-adherent. Of particular interest are convergences that are complete with respect to a collection of convergence covers. The following example goes back to the origins of completeness. Example XXIII.1.4. A metric d is complete if each d-Cauchy filter is td-adherent. By Proposition X.4.3, td constitutes P -complete if and only if d constitutes a complete metric. By Lemma X.4.1, a td-adherent d-Cauchy filter is td-convergent. Example XXIII.1.5. A convergence is compact if and only if it is ∅-complete. Here ∅ stands for the empty collection of families of subsets of X. Therefore, every filter F on X is ∅-fundamental, because there is no P in ∅. In other terms, ξ is ∅-complete whenever adhξ F 6= ∅ for each filter F. Let P and Q be collections of families of subsets of X.

Lemma XXIII.1.6. If F is Q-fundamental, and for every P ∈ P , there exists a refinement Q ∈ Q of P, then F is P -fundamental.

Proof. Let P ∈ P , and let QP ∈ Q be a refinement of P. As F is Q-fundamental, there exists QP ∈ F ∩ QP , hence there exists P ∈ P such that QP ⊂ P , and thus P ∈ F. 

Corollary XXIII.1.7. If a convergence ξ is P -complete, and for every P ∈ P , there exists a refinement Q ∈ Q of P, then ξ is Q-complete.

Proof. If F is Q-fundamental, then by Lemma XXIII.1.6, F is P -fundamental, hence adhξ F = 6 ∅ by P -completeness of ξ. Thus Q-completeness follows.  If P is a collection of families of sets, then let P ∪↓ := {P∪↓ : P ∈ P }.

Corollary XXIII.1.8. An ultrafilter U is P -fundamental if and only if it is P ∪↓ -fundamental.

Proof. Necessity follows from Lemma XXIII.1.6. The converse holds, because if an ultrafilter contains a finite union, then it contains at least one of its components. 

558

XXIII. COMPLETENESS

Actually, it is enough to consider completeness with respect to collections of ideals. Proposition XXIII.1.9. A convergence is P -complete if and only if it is P ∪↓ -complete. Proof. If ξ is P -complete, and a filter F is P ∪↓ -fundamental, then each U ∈ βF is P -fundamental by Corollary XXIII.1.8. Therefore, F is ξ-compactoid, so that ξ is P ∪↓ -complete. Conversely, if ξ is P ∪↓ -complete, and F is P -fundamental, then by Lemma XXIII.1.7, F is P ∪↓ -fundamental, because P is a refinement of P ∪↓ .  Recollect Definition XIX.3.14 that S is a ξ-regular refinement of P if adh\ξ S is a refinement of P. We have already observed in XIX.3.15 that Lemma XXIII.1.10. Each cover of a regular convergence has a regular covering refinement. XXIII.2. Cocompleteness Let H be a collection of families of subsets of X. Definition XXIII.2.1. A convergence ξ on X is called H-cocomplete if adhξ F 6= ∅ for each filter F such that F⊥H for each H ∈ H. By our convention,

(XXIII.2.1)

Hc := {H c : H ∈ H},

H ¬ := {Hc : H ∈ H}.

Proposition XXIII.2.2. A convergence is H-cocomplete if and only if it is H ¬ -complete. Proof. By definition, a convergence ξ is H-cocomplete if and only if adhξ F = 6 ∅, provided that F does not mesh any H ∈ H. Now, F⊥H means that there exists H ∈ H such that H c ∈ F, equivalently F ∩ Hc 6= ∅. 

Consequently, completeness with respect to a collection P is tantamount to cocompleteness with respect to the collection of the complements of families from P . Both formulations can be useful, as one or another becomes more incisive in different situations. Because by (XVIII.3.8), adhSξ G = adhξ G, a convergence ξ is H-cocomplete if and only if Sξ is H-cocomplete. Definition XXIII.2.3. A filter F is called H-cofundamental if F⊥H for each H ∈ H.

XXIII.2. COCOMPLETENESS

559

Proposition XXIII.2.4. A filter F is H-cofundamental if and only if it is H ¬ -fundamental. Proof. Indeed, by definition H-cofundamental whenever F⊥H for each H ∈ H, equivalently if F ∩ Hc 6= ∅ for each H ∈ H.  It follows immediately that

Proposition XXIII.2.5. A convergence ξ is H-cocomplete if and only if one of the following equivalent conditions holds: (1) each H-cofundamental filter is ξ-adherent, (2) each H-cofundamental ultrafilter is ξ-convergent, (3) each H-cofundamental filter is ξ-compactoid. Proof. Each ultrafilter is a filter, and if an ultrafilter is adherent, then it is convergent. On the other hand, a filter is compactoid whenever each finer ultrafilter is convergent.  By Propositions XXIII.1.9 and XXIII.2.4, cocompleteness with respect to a collection of families with the finite intersection property is equivalent to cocompleteness with respect to the corresponding filters. If H is a collection of families of sets, then let H ∩↑ := {H∩↑ : H ∈ H}.

Corollary XXIII.2.6. A convergence is H-cocomplete if and only if it is H ∩↑ -cocomplete. Notice that both the definitions of H-cocompleteness and of H-compactness follow a similar general pattern. Let CH stand for the family of H-cofundamental filters on a set X. Then, in view of Definition XX.3.1, Proposition XXIII.2.7. A convergence is H-cocomplete if and only if it is CH -compact Accordingly, H-cocomplete and H-compact convergences coincide if CH = HX. For instance, C∅ = FX, so that ∅-complete and compact convergences coincide. By Lemma VIII.2.3, a family P is a ξ-cover if and only if adhξ Pc = ∅. Therefore, Corollary XXIII.2.8. A convergence ξ is P -complete and P is a ξ-cover if and only if ξ is P ¬ -cocomplete and adhξ H = ∅ for each H ∈ P ¬.

Recapitulating, the same situations can be described either in terms of fundamental filters and completeness, or of cofundamental filters and cocompleteness. The former language is that traditionally used

560

XXIII. COMPLETENESS

in topology. The latter has been introduced recently (2), and has an advantage of enabling representations in terms of spaces of ultrafilters. Namely, Definition XXIII.2.3 can be reformulated in terms ultrafilters, as follows Lemma XXIII.2.9. A filter F is H-cofundamental whenever [ βF ∩ βH = ∅. H∈H

Therefore,

Corollary XXIII.2.10. A convergence ξ on X is H-cocomplete if and only if [ βX \ ξ − (X) ⊂ βH. H∈H

If, moreover, H is composed of non-ξ-adherent filters, then [ βX \ ξ − (X) = βH. H∈H

A convergence ξ is H-cocomplete if and only if all the ultrafilters of βF are ξ-convergent, provided that F is H-cofundamental. Equivalently, if limξ U = ∅ for an ultrafilter U, then there exists H ∈ H such that U ∈ βH.

Corollary XXIII.2.11. If ξ is a convergence on X, and H is a collection of non-ξ-adherent filters, then ξ is H-cocomplete if and only if {βH : H ∈ H} is a β-compact covering of βX \ ξ − (X). By Corollary VIII.1.10,

Lemma XXIII.2.12. If f ∈ C (ξ, τ ), then

adhξ H 6= ∅ =⇒ adhτ f [H] 6= ∅.

Corollary XXIII.2.13. If ξ is H-cocomplete and f ∈ C (ξ, τ ), then τ is {f [H] : H ∈ H}-cocomplete.

Proof. Let G⊥f [H], equivalently f − [G]⊥H, for each H ∈ H. As ξ is H-cocomplete, adhξ f − [G] 6= ∅, thus by Lemma XXIII.2.12, ∅ 6= adhτ f [f − [G]] ⊂ adhτ G.  XXIII.3. Finitely complete convergences In this section we study in detail the case of finite completeness. Be aware that the results presented here constitute a special case of general facts discussed in Sections XXIII.5, XXIII.6 and XXIII.7. Because of Proposition XXIII.1.9 and Corollary XXIII.2.6, completeness can be defined either in terms of collections of ideals or of filters. 2See

[60].

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