Convergence and Uniformity in Topology. (AM-2), Volume 2 9781400882199

Table of contents :
INTRODUCTION
CONTENTS
General Usage of the Alphabets
Special Usage of the Alphabets
Usage of General Symbols
Chapter I Ordering
Chapter II Direction
Chapter III Convergence
Chapter IV Compactness
Chapter V Normality
Chapter VI Structs
Chapter VII Function-Spaces
Chapter VIII Examples
Chapter IX Discussion
Bibliography
Indices

Citation preview

ANNALS OF MATHEMATICS STUDIES NUMBER 2

INTRODUCTION

This is an attempt to give a convenient, systematic, and natural treatment of some of the fundamentals of topology--the fundamentals of a topology which will be a tool of general applica­ tion to other branches of mathematics as well as a n 4interesting and satisfactory theory. In this treatment we show that, theoreti­ cally and practically, convergence is a notion of central importance in topology. The results set forth here are not all new; there are old results with old proofs, new results with old proofs, old results with new proofs, and new results with new proofs. In large part this material is drawn from my doctor’s thesis (Princeton University, 1939). I take this oppostunity to thank the many friends who have discussed various related questions with me. In particular, A. H. Stone, B. McMillan and B. Tuckerman have read the manuscript and made many helpful suggestions.

iii

CONTENTS

Page ...............

vii

General

Usage of

the Alphabets.

Special

Usage of

the A l p h a b e t s ..................vlii

Usage of General Chapter

I

Symbols.

.....................

ix

O r de r i n g .................. ..

I

Chapter

II

D irecti o n ......................

Chapter

III

Convergence

.................... 16

Chapter

IV

Compactness

....................

31

......................

43

Chapter V

Normality

Chapter VI

S t r u c t s ......................... 55

Chapter

VII

F u nct i on- S pa c es

Chapter

VIII

Examples.

Chapter

IX

D i s c u s s i o n .......................82

...........................

10

71

...................... 78

Bibl io g r a p h y ............................... ..

87

I n d i c e s ............................................... 89

v

GENERAL

USAGE

OF THE

ALPHABETS

Small Latin Letters: a, b, c, •••, are used for individuals (indices, elements, points, etc.). Latin capitals: sets, collections, etc.

A, B, C, •••, are used for

German capitals: S, §, £, 2Jt, 51, U, 55, ffi, are used for collections of sets, in particular for coverings. Script capitals: G , B, C, •••, are used for ordered systems. Except in Chapter I, these ordered systems are assumed to be directed. Small Greek letters: a, ••• v, are used for finite sets. used for functions.

p , y, cp and

6 , e, y are

Greek capitals: T 9 A, are used for collections of finite sets, when these finite sets are denoted by small Greek letters. Dashed small Greek letters: a, fj, y, B, •••, are used for sets which are regarded as elements and which need not be finite. Dashed Greek capitals: T, A, •••, are used for collections of sets denoted by the corresponding dashed small letter. (An exception to the last four rules occurs in Chapter V I I I . )

vii

SP E C I A L

U SAGE

OF THE

ALPHABETS

Corresponding letters in different alphabets are used for related objects. D is frequently used to denote 2X ered as a set rather than as a set of sets. The indices i, j, k, 1, m, and run thru the positive integers.

n

consid­ usually

N is used frequently in two distinct senses; as the set of all positive integers, or as anbd of the point being considered. U , V , and tt , S3 , and

W

are open sets.

8 are open coverings.

These conventions are usually independent of indices. Examples. a x is usually an element of A . y 1 is u su­ ally a finite subset of C . 5 belongs to A (or perhaps to A x c A ).

viii

USAGE

OF

GENERAL

SYMBOLS

e

denotes class-membership,

c

and

n and tion and union.

z? denote set-theoretical

inclusion.

u denote set-theoretical

intersec­

< , > , a and lattice operations.

v

denote the corresponding

\ } are used to form sets. Thus, is the family of all U's. Also ix'} is often the set made up of x 1 alone. I is used to indicate domains, etc. It can usually he read "where” or ’ ’ running t h r u ”. Thus f(x|X) is a function defined for x in X , and {f(x)|xeX} , often written |f(x)|X} is the set of its functional values.

0

denotes the empty set.

Propositions whose proof is left to the reader are designated by rather than by words "Lemmatf or "Theorem".

ix

the

CONVERGENCE AND UNIFORMITY IN TOPOLOGY BY

JOHN W. TUKEY

PRINCETON PRINCETON UNIVERSITY PRESS LONDON: HUMPHREY MILFORD OXFORD UNIVERSITY PRESS

I9 4 O

Copyright 1940 PRINCET ON UNIVERSITY PRESS

P r i n t e d i n u . s .a .

Lithoprinted by Edwards Brothers, Inc., Lithoprinters Ann Arbor, Michigan, 1940

Chapter

I

ORDERING

1. Introduction 2. Set Theory 3. Orderings 4. Functions 5. Products 6. Z o r n ’s Lemma 7. Cardinal

Numbers

1. Introduction. In this chapter we concern ourselves chiefly with definitions and notation for sets, functions, and ordered systems. We assume familiarity only with the most elementary facts about sets and functions. Definitions and notation must represent a compromise b e ­ tween theory and practice. We tend to use weaker definitions and more precise notation than some authors. In places we fol­ low the definitions and notation used by N. Bourbaki and his collaborators. Certain conventions of notation, which allow the nota­ tion to carry implicit information, and which are used consis­ tently thruout, are laid down in this chapter. We begin with our definitions and notation for set theory (§2), ordered systems (§3)> and functions (§4). These are applied to products of ordered systems (§5) and Zorn's Lemma (§6 ). We conclude with a modicum of the theory of cardinal numbers (§ 7 ). 2. Set Theory. We assume that the reader is familiar with the elements of set theory, and concern ourselves with notation. x|P . tt,a IA

The set of elements x which have property P is This is often further abbreviated in obvious ways, ad for tta | a e A .

This last abbreviation is an example of an important convention. We use small letters, a , b , c , ••• for points belonging to sets denoted by the corresponding capital letters, A , B , C , ••• . For example, if B is a subset of a set A of real numbers,> then ’’For every a and some b , b > a 11 means "For every element a of A and for some ele­ ment b of B (which may depend on a , otherwise we would say "for some b and every a ") b > a ".

2

ORDERING

We often use the letters i , j , k , 1, m , for integers and N for the set of all positive integers.

n ,

We consider a set, its points, and its subsets. If the point a belongs to the set A we write aeA or Asa . (We have no hesitation in inverting any relation symbol.) If every point of A belongs to B we write AcB or B d A and say that A is a subset of B . The empty set is 0 (pronounced as the phonetic symbol and Scandinavian vowel). We make no formal dis­ tinction between the empty subsets of different sets. The points belonging to either A or B or both form the union AuB of A and B . (We have adopted n , u, "intersection” and "union” rather than the older + , • , 2 , U , "prod­ uct" , and "sum" to avoid confusion when the sets belong to an algebraic entity (group, linear space, etc.). The points common to A and B form the Intersection AnB of A and B . The union of a collection of sets Ah|H is denoted variously as U h A h , UfAhlH} , and UhAhlH . The inter­ section of AhlH is treated similarly. Two particular cases are U{Ahl0> = 0 and fUA^I)#} = X , where X is the set, with whose subsets we are operating. The points in A but not in B form the set A-B (This notation is not very satisfactory). The set of all sub­ sets of C is 2 c . If AnB ^ 0 we say that A meets B ; if AnB = 0 we say that A and B are disjoint.A collection of sets disjoint if each pair of them is disjoint.

is

We will often use alternative terms for "point" , "set" , and "subset" , in an effort to avoid confusion. E l e ­ ments, indices, objects, etc., may belong to or be contained in families, classes, systems, aggregates, or collections. The prefix sub is used in the same sense when applied to all these terms. A typical usage is, "The system of all families of sub­ sets of A ." If, on the other hand, we apply the prefix sub to a word meaning a set with an attached "structure" of some sort, e.g., "space" , "ordered system" , "group" , we imply that the set concerned is a subset with the "structure" induced by the "structure" of the object of which it is a subset. 5. Orderings. We consider a set A and a binary relation > defined on the set (more precisely, perhaps, defined on the product of the set by itself). That Is, for each ordered pair a* , a" of elements of A , one of the incompatible formulas a* > a" or a 1 / a" is accepted. (We denote negation by can­ cellation with / ). If the relation is transitive, that is, if we have

1-3.1 ... 1-3.2 3.1

a?

> a"

and

a"> a*

imply

3

a f > a* ,

then we say that G = (A,>) is an ordered system. (We use ‘’element" and "system" when discussing sets with an ordering relation.) An ordered system may have several important properties. It is linear if either a* > a" or a ’ < a" or both, for every pair of distinct elements a' , a" . It is properly ordered if a' > a" and a* < a" imply a* =a" (that is, a ’ and a" are the same). It is reflexive if a > a for all a . It is lrreflexive if a / a for all a . It is symmetric if a* > a" implies a" > a ’ . It is trivially ordered if a 1 > a" for all a ’ and a" . It is vacuously ordered if a 1 / a" for all a' and a" . If B is a subset of A , and > is defined on A , then it is defined on B ; if > is transitive on A , then it is transitive on B . Hence, if G = (A,>) is an ordered sys­ tem, then B = (B,>) is an ordered system. We call B a sub­ system of G , implying thereby that the ordering relation of B can be obtained from that of G in this way. Two ordered systems are isomorphic if there is a one-toone correspondance between them which carries their order rela­ tions into each other. We write G - B for this relation. It is clear that all the notions discussed in this § are invari­ ant under passage to an isomorphic system. Associated with each set there are two ordered systems: the vacuously ordered system made up of the points of the set, and the system of all subsets of the set ordered by id . There is a logical distinction between a set and a vacuously ordered system, but we shall have no occasion to insist upon it. The second ordered system and its subsystems are of ex­ treme importance. If we speak of a system of subsets of some set, we will assume, unless we specifically indicate the con­ trary, that the system is ordered by zd . We denote a subset of C by y and a collection of y's by T . ( T is usually not the collection of all subsets of C , for which we have already introduced the notation 2 C ). A set containing a finite (or zero) number of points is a finite set. ( 0 is a finite set.) A finite subset of C will be denoted by y , and the collection of all finite subsets of C will be denoted by T . We apply this correspondance between the different alphabets to other letters. We note that every system of subsets is reflexive and properly ordered. 3.2 Lemma. Every reflexive and properly ordered system morp hi c to a system of subsets of some set.

is iso­

4

ORDERING

P ro o f . Let C = (C,>) be a reflexive and properly ordered sys­ tem. To c' we let correspond y ( c ' ) = {c|c < c ’} . The reader may easily show that this is a one-to-one correspondance between C and a collection T of subsets of C (here T will never consist of all the subsets of C ), and that this correspondance takes > into zd and into > . It is well known that an equivalence relation (that Is, one which generates a reflexive, transitive and symmetric ordered system) divide the set on which it is defined into mutually exclusive equivalence classes. If we denote the equivalence relation by ~ and the equivalence class containing a by [a] , then b e [a] if and only If b ~ a. We remark that every ordered system has a natural equivalence relation de­ fined by 3:3

a T~ a"

if either

a T = a"

or both

a 1 > a"

and

a n
[b] if a > b . The reader can easily show that this definition is Independent of the particular representatives a and b of the classes [a] and [b] . 3,1 ****, Such an ordered system of eq ui valence classes properly ordered.

from

G

is

We use [CL] for the properly ordered system derived In this way,

3.5 ****.

|f

G

is reflexive,

then so is

[G] , and conversely

So far, except foo? systems of subsets and equivalence relations, we have used > as the symbol of the ordering rela­ tion in which we were interested. There is, of course, no necessity for this, and we will use several other symbols for particular relations. However, the usage of terms we are about to discuss is usually restricted to ordered systems whose order­ ing relation is written > or ^ . If a > b for all b e B , then a is an upper bound of B . We define lower bound similarly. Any of the upper bounds of B which is a lower bound of the class of all upper bounds of B Is a supremum (sometimes called "least upper bound” or "join" ) of B . A given B may have no supremum, one supremum, or several suprema. When we say " a is a supremum of B ” , we should, to be precise, say n a is a supremum in G of B " , but it will usually be clear what or­ dered system Is meant. If G is properly ordered, then* B c G has at most one supremum. An lnflmum ( ’’greatest lower b ound” or ’’m e e t ” ) is defined similarly and is subject to simular re ­ marks. We use the symbols v and a for suprema and infiraa. Thus, a ’v a ” is a supremum of a' and a ” ). In particular

1-3.5

...

5

1-4

cases (for an example see V-2), where the system is not proper­ ly ordered, we may use some definite way to define a'va" as a particular supremum of a ’ and a ” . We denote the supremum of {btlT} variously as: Vtbt , V{btlT} , VffbtlT} • We use A for the infimum in a similar way. We depart from these notations if the elements are real numbers and the relation is ’’greater than or equal t o ” in its elementary sense, when we use ”Sup" and ”I nf” instead of V and A . If each pair of elements (not necessarily distinct) has at least one supremum and at least one infimum (the pair a , a need have a neither as a supremum nor as an infimum), the ordered system is a trellis. If every subsystem has at least one supremum and one infimum, then the trellis is a complete t rellis. The reader familiar with the definition of a lattice will see that a lattice is a reflexive and properly ordered trellis. Hence (3*5) the ordered system of the equivalence classes of a reflexive trellis is a lattice. We suggest the consideration of the following ordered systems (and, possibly, some of their naturally arising sub­ systems); they should serve to make the meaning of these defini­ tions clear. 1) The elements are the positive integers, or the posi­ tive and negative integers, with one of the relations: > (definitely greater than), _> (greater than or equal to), \ (divides), = (mod p) (congruent modulo p). 2) The elements are all finite sets of real numbers with one of the following relations: > T meaning ’’contains more points than” , >” meaning ’’contains at least as many points a s ” , > f” meaning ’’contains exactly as many points as” . 3) The elements are all measurable subsets of closed interval [0 ,1 ] , or of the line (-°s+°©) ,with A >° B meaning meas(B-A) = 0 . 4) The elements are a , b , and c , tion holds in the following cases only: a > a , c > a , c > b , c > c . 5)

Que r y :

the

and the rela­ b > a ,

Is any trivially ordered system a trellis?

4. Functions. We have no occasion to deal with functions which are not single-valued. Hence ’’function” will be understood to mean "single-valued function” . We shall indicate the domain of definition of a function in the notation for the function. A function defined on the points of X is, for exam­ ple, f(x|X) . f(x) is the value of f at x ; we never use f(x) to refer to a function. With certain exceptions, such as f as a function defined on X and having values in Y , the

6

ORDERING

letter denoting the function is often associated with the set in which its values lie. Thus a(t>|B) is a function defined for b in B , and is likely to have its values in A , or perhaps in A" or A a . We remind the reader of a few simple facts about such functions. If f is defined for x in X and has values in Y , then there Is a function defined for H In 2X and having Its values in 2 Y , which is also denoted by f and is defined by f ( H ) = { y l y = f ( x ) , xeH } . This process can be, and Is, continued to define a function from 2 2X to 2 2 Y , and so on. The function f _1 from 2 Y 2 X is defined by

to

f ~1(K) = { x | f ( x )e = K j. Aga^.n, we may repeat this process to define a function from 22 to 2 2 , and so on. We make this transfer from pointfunction to set-function or collection-function freely at all times. We have f(f-l(H)) = H , but only f - 1 (f(H))=>H . f(a|A) Is an extension of f(h) = g(b) for each b e B .

Hr

g(b|B) Is B c A

and

The notation distinguishes between f(g(x|X)) and f(g(x) IX) . When we write f(g(xJX)) we imply that g is a function defined on X , and that, as a point, g(xiX) belongs to the domain of definition of f , whose value at g(x|X) f(g(x|X)) . When we write f(g(X) |X) we imply that g is a function defined (at least) on X , and that g(X) is a part of the domain of definition of f ; in these circumstances, f(g(X) |X) is a function defined on X , whose value at x is f(g(x)) . 5. Products. The formulation of the product of ordered sets that we now give may seem strange at first, but I feel that it is the natural and convenient one. The product of two simple sequences is usually described as the set of pairs (m 1, m") where m' and m n are positive integers and (mf,m") > (nf,n” ) if both m* > n 1 and m M > n ” . We prefer to regard the ele­ ments of the product as functions defined for the points 1 and 2 with functional values in a simple sequence. This is the notion that we are going to generalize. If, for each c e C , d c = (Ac ,>°) Is an ordered sys­ tem, and if T is a system of subsets of C , then we consider all the functions a(c|y) , where yep and a(c)*Ac for all cey . We define > by: a ' f c l y 1) > a'T(c|yM ) if for each cey” we have cey' (hence y 1 zd yft ) and a 1 (c) >c a ”(c) . The re­ sulting ordered system of functions is called the product of the

is

1-5 ... 1-6.3

7

GP by r , and Is written (r,flc|C). The systems (Xc are factors of (T,ftc |C). If there is n o ^ e T s u c h that cs ?7 then Clc is an Inessen­ tial factor. In the example above, C = [1,2], r consists of a single subset of C , namely C itself, and 6 1 and (X2 are each the sequence of positive integers in their natural order. If f® consists of C alone, then we write pca c p[gc |c] , or pc iac |G] for (r,ac IC ). We call such a product the product of the 0.° . This is, of course, the classical case. Thus the example above is the product of two simple sequences. A product of ordered systems is ordered. A product is properly ordered, vacuously ordered, reflexive, or irreflexive, if all its essential factors have the corresponding property. If [Ac |C] is a collection of sets, then we may regard the sets as vacuously ordered systems and form Pl&c |C] . This product is a vacuously ordered system and may be considered a set. Thi 3 set is called the product of the sets A . There is a slight further generalization of this notion of product, which we shall not need, and which we shall omit.

6 . Z o r n ’s L e m m a . We take an important and convenient lemma due to Max Zorn as an axiom. It is equivalent to the axiom of choice, and is a more convenient and powerful tool than transfinite induction. It also has the advantage that It can be easily stated in elementary terms. An element a ’ of the ordered system (A,>) is maximal if a > a ’ implies a'>a . (This coincides with the usual defini­ tion for a properly and linearly ordered system.) 6.1 Zorn*s Lemma (First Form). An ordered system, each of whose 1 inear subsystems has an upper bound, contains a maximal e 1e m e n t . 6.2 Z o r n ^ Lemma (Second Form). Every ordered system contains a linear subsystem B # such that every upper bound of B (if any ex is t) belongs to B.

A property of sets is of finite character, if a set has the property when and only when all its finite subsets have the property. 6.3 Zorn*s Lemma (Third Form). Given a set and a property of finite character, there exists a maximal subset having the property.

A condition on a function is a finite restriction, if, as a property of the graph of the function, It is of finite char­ acter. This means that it is the logical sum of conditions each

8

ORDERING

of which depends on the functional values at a finite number of points. Some examples of finite restrictions are: that the function-1) be a constant. 2 ) vanish on a certain set. 3 ) be additive (if defined on a group, linear space, e t c .) 4) be bounded in absolute value by a particular func­ tion . 6.4 Zorn's Lemma (Fourth Form). The class of those fu nctions defined on subsets of a given set and satisfying a given family of finite restrictions, contains a function no one of whose extentions belongs to the class.

We suggest some results and methods of proof; the reader is advised to complete the proofs. 1) 6.1 implies that, if A and B are sets, there is either a one-to-one mapping of A onto a subset of B or a one-to-one mapping of B onto a subset of A . (That is, the cardinal numbers of A and B are comparable.) For the rela­ tion of extension orders the system of all one-to-one mappings of a subset of A onto a subset of B .

2 ) 6.3 implies that any set of numbers contains a m axi­ mal set algebraically independent over a given field. For the property of being algebraically independent over a given field is of finite character. 3) 6.4 Implies the Hahn-Banach theorem (Banach 1932, pp. 28-29). For the properties of being additive, homogeneous, and bounded by p , are finite restrictions. 4) 6.1 implies 6.2. For the system of all linear sub­ systems of a given system, when ordered by inclusion, satisfies the hypothesis of 6 .1 . 5) 6.1 implies 6 .3 . For the system of subsets of the given set having the given property of finite character, when ordered by inclusion, satisfies the hypothesis of 6 .1 .

6 ) 6.2 implies 6.1. For an upper bound for the linear subsystem o f 6.2 is clearly maximal. 7) 6.3 implies 6.2. For the property of subsystem is a property of finite character.

8) functions.

6.3

implies 6.4.

being a

linear

For consider the graphs of the

9) 6.4 implies 6 .3 . For the property of being a char­ acteristic function is a finite restriction, and a property of finite character for subsets is a finite restriction for the corresponding characteristic functions.

1-7

9

7. Cardinal Num b e r s . We need a very little of the theory of cardinal numbers, which we now recall. Two sets have the same cardinal number (or pote n c y ), if there exists a one-to-one cor­ respondence between them. (Compare l) above.) The cardinal number of A is denoted by IAI . We write IAI - IBI if there is a one-to-one mapping of A on a subset of B . A set is countably infinite if it has the same cardinal number ( ) as the set of positive Integers. A set is count­ able if it is finite or countably Infinite. The union of a countable family of countable sets is countable. The product of a finite collection of countable sets Is countable. A ny subset of a countable set is countable.

Chapter

II

DIRECTION 1 . Introd uct io n . 2. Directed Systems. 3. An Order in g . 4. Systems of Subsets. 5. Stacks. 6. The Countable Case. 7. The General Case.

1. Introduction. In this chapter we are concerned with the theory of directed systems for its topological applications. We discuss, therefore, only a part of the theory; a fuller account will appear soon in another place. 2. Directed Systems. A directed system is a non-empty ordered system, d = (A,>) , in which we have 2.1 For every a f and a", and some a*, a * > a f, a*>a".

This property was originally called "the composition property” (Moore and Smith 1922). We note that 2.1 may hold for (A,>) and not for (A,) , Tfr = (B,>) , etc. We often use (X where A should strictly be used, as a «= d for aeA. Three characteristic examples of directed systems are: 1° Any linearly ordered set. 2° A system of subsets of some set, directed by ; for example, the system of all finite subsets of a set. 3° The neighborhoods of a given point in a space, which are directed by c . There are two Important classes of subsystems of a d i ­ rected system; the cofinal subsystems and the residual subsys­ tems. A subsystem "B of a directed system (X is cofinal in G if 2.2 For each a, and some b y

^

is residual in d

b>a.

if

10

II-2.3 ... II-2.11 2.3 For some a', and all

a>a ’ ,

11

aeft.

These concepts are related by 2.4 in U .

is cofinal i n C if and only if G',(x > d ft > TV . If a ' s f i ’ , then, since IS1 i s cofinal in C' , we may choose b ' e Ti' so that b ’ > a 1 . Hence we may choose b'fa'lA1) so that b ' ( a 1 ) > a' for each a' ; similarly we may choose a ’fb'lB') sothat a ' ( b 1 ) >t>' for each bf . Hence (3 -3 ) d' > TV and ft’> G f , so that (3.2) G > 13 and ft > G Suppose G > ft and ft > G . We assume that AnB = 0 , which can be obtained by considering a system isomorphic to ft . There exist (3*3) a(blB) and b(alA) such that, a > a(b) implies b(a) > b, and b > b(a) implies a(b) > a . We now extend the definition of > to D = AuB by setting a > b, b > a,

if if

a > a(b) ; b > b(a) .

It is clear that J9 is directed by are cofinal in J0 . Hence G~7S.

>

and that

G

and ft

3.5 ****. G~ft if and only if there exist functions a (b | B ) and b ( a | A ) such that, a>a(b) implies b(a)>b, and b>b(a) implies a (b)> a .

,

f

15

11-3-6 ... II-6 The interested reader might prove 3.6 *♦**.

If C is finite,

then

P[AC 1C1

is a supremum of

£A * |C ] .

While we shall not need this result, .It has consideraole intrin­ sic interest, since it shows that > directs the class of "all" directed systems. 4. Systems of S ubsets. We shall see that each cofinal type con­ tains at least one system of subsets. We use the methods of 1-3 • 4.1 Lemma. I f (X is d i r e c t e d , then so is systems are c o f i n a l l y s i m i l a r .

[&] , and these two

P r o o f . We apply 3*5* where (setting T2> = [ (X ]) a(b) is some representative of b = [a] , and b(a) = [a] . We leave the details to the reader. 4.2 Lemma. l f d = ( A , > ) is d i r e c t e d and i f a' = ( A , 0 . where a ' u " i f e i t h e r a ’ >a" or a ' a f t " ; then (X1 is d i r e c t e d , and & and a , let , and let b(a) = a* (a) and a(b) = a'(a) , where then the lemma follows from 3 *5 *

***♦. Two isomorphic d i r e c t e d

systems are c o f i n a l l y

sim ilar.

Prom 4.1, 4.2, 4.3, 1-3*2 and 1-3.5 we see immediately that 4.4 * * * * . subsets.

Each c o f i n a l

t ype c o n t a i n s

at

least

one system of

For many Interesting problems, which lie outside the scope of the present discussion, we could restrict ourselves to those systems of subsets which are Ideals in the ring of all subsets of a set. 5. Stacks. A stack (this term is due to M. M. Day) is the sys­ tem of all finite subsets of some set, called the base of the. stack. We recall that T denotes the stack with base C , etc. The topological importance of stacks is related 5.1 Theorem.

For any d i r e c t e d

system^,

to

we haveA>45 .

Pr o o f . Let , then since 6 is finite, there exists su d( 6 ) e such that d( 6 ) > d for all d e 6 . If we let 6 (d) = d (the set made up of d alone), then 636(d) means d e 6 which Implies d(6 ) > d . Hence 3*1 holds, and A>J0.

6 . The Countable C a s e . Those directed systems which have at most a countable number of elements are of Interest. Clearly,

14

DIRECTION

every directed system containing a maximal element is cofinally similar to every other such system. If G is a directed system without maximal element containing at most a countable number of elements, then we may suppose that £an |N] is an enumeration of the system. Set b x = a x , and choose b n so that bn > a n and b n > b . The subsystem B is obviously cofinal in (X . If, for some n f , we had bn < b n i for all n > n' , then b n » would be a maximal element of TJ and hence of d , con­ trary to hypothesis. Since this cannot happen, we can choose a subsystem C = [ c k = b n k |Kj of T5 , such that c^ > and Ck / Ck-Z for all k . It is clear that C is cofinal in T3 and hence in G and it is also clear that C is iso­ morphic to the system of positive integers In their natural order, that C is, in fact, a simple sequence. Hence we have proved 6.1 Theorem. A directed system with a countable number of elements contains either a maximal (that is, cofinal) element or a cofinal simple sequence. In particular, every such system belongs to one of two cofinal types, the one containing the finite directed system or the one containing the simple seq u e n c e .

7. The General C a s e . The simplicity of the countable case is absent in the general case. Tho there remain many unsolved prob­ lems, we can give some idea of the situation. We shall omit all semblance of proof. There Is a family of easily accesible cofinal types. The structure of the lower part of this family with regard to > is

u

A 04

00

£.0 1 A

11

L-

22

Here the indicated relations generate all those which exist by transitivity (thus 02 > 12 > 11 implies that 02 > 11 ; and it is not true that 01 > 22). All these cofinal types are > the type containing all finite directed systems. The types 00 , 01 , 02 , etc. contain stacks. The types 11 , 22 , 33 t etc. contain transfinite sequences. We see that "transfinite sequence > stack” never occurs. The co­ final similarity between simple sequence and stack on a count­ able base Is purely the result of countability and does not generalize.

II-7

15

Prom this family we can generate a complete lattice of cofinal types. However it is not known whether the system of all cofinal types (suitably restricted to avoid paradoxes) is a complete lattice or not. Neither Is It known whether each pair of cofinal types has an infimum or not.

Chapter

III

CONVERGENCE 1.

1ntrod uct ion.

2. Co nv ergence and Phalanxes 3. The Bill

of R ights.

4. Effect iv en e s s . S. Relat iv izat ion. 6. Open Sets and T-spaces. 7. Cont inu it y . a. Separa ti on Axioms. 9. H istor ical R e m a r k s .

1. Introduction. The aim of this chapter is twofold: to estab­ lish convergence as a basic concept equivalent, in a very wide class of spaces, to closure and neighborhoods; and to make con­ vergence available as a tool In spaces where the basic concept is closure or neighborhoods. We have also collected here some other topological results of an elementary nature which we will need later. In §2 we discuss the applicability of the terra "conver­ gence" and point out the phalanx as particularly useful. In § 3 we establish the equivalence of closure, convergence and neigh­ borhoods under very general conditions. In §4 we discuss the effectiveness of the various directed systems as carriers of convergence. In § § 5 and 6 we discuss relativization and open sets. In §§7 and 8 we deal briefly with continuity and the separation axioms. In § 9 we make some historical remarks. The results of this chapter may be summarized as follows: The term "convergent" concerns functions defined on directed systems. Convergence is equivalent to closure and neighborhoods in all reasonable spaces. When we use convergence in a general problem we need consider only phalanxes. 2. Convergence and Phalanxes. While the notion of convergence can be further generalized (cp. Garrett Birkhoff 1939 ), I feel that, for the main purposes of topology, the convenient class of objects, whose "convergence" is to be considered, is the class of functions defined on a directed system and taking values in a space. All the further generalizations that I know which

16

Ill-2 ... Ill-3.la

17

preserve uniqueness of limit in the classical spaces can be re­ duced to this notion. This was the attitude taken by Moore and Smith in 1922 when they discussed the convergence of real-valued objects. Classically, one considered sequences; a sequence is pre­ cisely a function on the directed system of the integers. Stacks (II-5) play a special rected systems; functions defined on in problems of convergence. We call The base of the stack is the base of with base A is an A-phalanx.

role in the theory of d i ­ stacks play a special role such a function a ph a l a n x . the phalanx. A phalanx

We shall modify our functional notation when it is con­ venient. A function on the directed system d will be denoted by x(a|G) as well as x(a|A) In order to refer to the order­ ing relation of d . If CL Is the stack T , we write x(*y|C) rather than x(^ |r). One reason for the Importance of the phalanx is its ease of manipulation. A phalanx x(a|A) is inflated to a phalanx Xj^^u^IAuB) by defining x x (aup) = x(«). (Here A and B are disjoint.) This essentially introduces dummy elements and is important for results about the selection of suitable subphalanx­ es. Two phalanxes x(cc |A ) and y ((51B) are meshed to a phalanx z(y |C) , where C = AuB, when x(a|A),

(

y(|i|B),

if

|y| is odd,

if

\y\ is even.

3. The Bill of Rig h t s . We now set forth the relations between closure, convergence and neighborhoods in very general spaces. We are dealing with a certain set X ,Its subsets and points. A closure operator assigns to each subset H of X a closure H . We shall usually assume 3.1 9 *0 , • HUK=HoK, of which an easy consequence is 3.1a HcK

implies HcK

(but we shall not assume, for the present, more special rela­ tions, such as H = H, If = H ) . A notion of convergence detertermines, for each function x(a| a such that x ( a ’)^N . That Is, d 1 = {a|x(a)eNl is cofinal In G. It is clear from this and 3.9 that the other half of 3 .1 1 holds. We proceed to the proof of 3.4. If x(a|£t) converges to x , and G f is cofinal in d , then (3 . H > 3 .1 3 ) x ( a | d ’) converges to x . Whence (3.9) x( applying 3.8 as well as 3.9. This completes the proof of 3.4. If xeH , and If IN] is a finite collection of nbds of *x , then (3«3) > for some nbd N f of x , NejNj Implies N'cN , and hence (3.8) H n N T ^ 0 . If 6esA , and If *N] consists of those nbds of x which belong to 6 , then we may choose x ’(6 )eH so that x ,(6 )^n{NJ . Hence, as in the proof

22

CONVERGENCE

of 3 . H > x'( 6 |D) converges to x . Thus one half of 3.7, 3.7P and 3.7PP is proved. If x(a|G) converges to x , and if x(G)cH , then ( 3 .9 ) , for each nbd N of x , HnN-Dx(G.)nN ^ 0 ; hence (3.8) xeH . This completes the proof o f 3 *7 , 3.7p and 3.7PP. H°. 3.10 and 3.4

imply 3.7, 3.7p and 3.7pp.

Let xeH . Let • {K| be the class of subsets of H such that xefl-K . If K* and K M belong to [Kj , then, since (3.10) H-(K'nK") = (H-K' )u(H-KM ) = H^K*uH^f1ir , x e H - ( K ’ K " ) and so K ’nl^’elKj . Thus an intersection of a finite number of K's is a K . Since xeH = E-0 , 0eK . Let 6e A ; let be the intersection of the K 1s belonging to 6 ; anQ. choose x 1 (6 )eK& . Then, if K e K and K e 6 , x ' ( 6 )eK . I say that x '(6 |D) converges to x . For if A'cz A is such that x e x ’ ( A’ ) , then H - x ’fA* ) is a K and no 6 containing this K can belong to A' . Hence A 1i s not cofinal in A . Hence, is A 1 is cofinal in A , then x e x 1 (Af ) . Conversely, let x(a|tf) converge to x and x((x)cH . Now (X is cofinal in CL , hence (3.*0 xex(&)eH . Hence 3»7, 3.7p and 3-7PP hold. 5*. 3.10 and 3.5

imply 3.8.

If xeH , and if N is a nbd of x ; then (3-5) xirfTN ; hence HtfX^N ; hence (3.10) Hcflt-N , that is NnH ^ 0 . Conversely, if, for all nbds, N , of x , HnN £ 0 , then, since Hn(X-H) = 0 , X - H is not a nbd of x . Hence (3.5) xeX-^X-H) = H . So we have proved 3.8. 6°. 3.11 and 3.6

imply 3.9.

Let x(a| 6i) converge to x , and let N be a nbd of x . Let = |a|x(a)^Nj . If were cofinal in Cl , then (3 .2 ) x(a|Ctf)would converge to x , which is impossible (3.6) since x(a» )nN = 0 . Hence (11-2.^) a-Gt 1 is residual in G. ; that is, there is an a 1 such that a > a 1 implies a^CL’ , that is, x(a)eN .Conversely, suppose that, for each nbd, N , of x , there is an a* such that a > a 1 implies x(a)eN . Let (2f be cofinal in G. If X-x(G* ) were a nbd of x ; then, for some a 1 , a > a f would imply x(a)eX-x(d’ ). Since (2' is cofinal in (X , this is impossible and X-x(Ct* ) is not a nbd of x . Hence (3.6) there is an x fl(b|B) such that (X-x(a» ))nxff(tt) = 0 , and x ” (b|T5) converges to x , Hence x" (T5)cx' (&f ) and (3.1l)> since B is cofinal in T5 , some x'( 6 |D) converges to x with x f (a)cx” (l3)cx(dT ) . Since a 1 was any cofinal subsystem of d , we have (3 .1 l) x(a|d) converging to x . So we have proved 3.9. 7°. 3.11 and 3.7

in £ .

imply 3.14.

Let x(a|d) converge to x , and let (X1 be cofinal Then (3.2)x(a|ft(in the

Proof. If Gi , there exist functions a(b| 13) and b(a|G) such that a > a(b) implies b(a) > b . If x(b|B) converges to x , then we set x ’(a) = x(b(a)) , thus defining x ’(a|6 ) . If N is a nbd of x , then (3.9 ), for some b 1 , b > b f implies x(b)eN . But if a > a ( b f) , we have b(a) > b' , hence x 1 (a)eN ; hence (3.9) x(a|G) converges to x . If G. is as effective as Ti , then we consider the following space; X = B u [o©j, where © o ^ b , and H is a nbd of o® if, for some b* , b > b* implies beH . The other points of X are assigned nbds in any way satisfying 3.12. The identi­ cal function (b*(b) = b) converges to « ; hence (4.1) some b"(a|G) converges to » . Now B(b) = | b 1 |b 1 > b] is a nbd of o o ; hence (3 .9 ) there is an a(b) such that a > a(b) implies b 11(a )eB(b ) , Hence d > . 4.3 Remarks. a) This theorem provides additional support for the use of > as a central relation In the theory of directed systems. b) There is (ll-5*l) & stack as effective as any particular directed system. Together with 4.2 this explains why phalanxes are sufficient for general topological purposes. c) In the countable case (II- 6 ) any directed system without last element is as effective as any other. This explains the usefulness of the simple sequence In the countable case. d) If x(n|N) Is a simple sequence, and if we set x ’(v) = x(|v|) , then x ’(v|N) is a phalanx which con­ verges to those points and only those points to which x(n|N) converges. The sequence and the phalanx have similar properties with regard to other notions (as "cluster point" IV-3) of a convergence nature. If x"(p|N) is a phalanx on a countable base (which we may as well take to be the set of the positive integers), and If we set v n = {n'ln 1 - n\ and x*(n) = x ,f(vn ) , then x*(n|N) converges to all the limits of x"(v|N) , but the converse need not be true. e) The similarity of sequence and phalanx where all is countable does not (II- 7 ) continue when we replace tf*by any uncountable cardinal number.

26

CONVERGENCE

5. Relatlvlzatlon. If X is a space, and topology of X induces a topology in Y . relations, 5.1 The closure

in Y, of H c Y ,

H (Y,# is equal to HnY.

5-2 x(a|&), where x((2)ciY c onver ges x(a|ft) converges to x in X. 5.3 The nbds of x e K nbd of x in X.

are all

YcX then the This occurs t h m the

in Y to

x e

Y if and only

sets of the fora YnN where

if

N is a

We have, 5.4 Theorem. If X is a space (3.16), then 5.1, 5.2 and 5.3 induce a topology in Y, making Y a space.

Proof.

We leave the proof to the interested reader.

Under the circumstances described In 5.4 we say that Y is a subspace of X , and that X Is a superspace of Y . We shall have occasion to use, 5.5 ****.

If xe=H and M is a nbd of x, then x e HTTn .

6 . Open sets and T-spaces. We begin with 6.1 ****. The following conditions on a subset U of a space X are equivalent; 6.2 jTUczX-U. 6.3

If x(a|&) converges to

6.3p 6.3pp

If xftylC) conver ges t o

x e

U, then x((X)nU«/0.

x e

U, then x(f~)nU^0.

If x(6|0), where D « 2 * f converges to xesU, then x(A)nU40.

6.4 U is a nbd of all

its points.

A set satisfying these equivalent conditions is op e n . Prom 6.4 we may easily prove, since every point has a nbd, that 6.5 ***•.

|n a space X open sets satisfy;

6.6 i and X are open. 6.7 The intersection of a finite number of open sets open set. 6.8 Any union of open sets

is -an

is an open set.

We shall be really Interested in open sets only when there are "enough” of them. The conditions for this are stated In 6.9 Lemma. For a space X, the following conditions are e q ui v a ­ lent; 6.10 The open sets contai ni ng x form a nbd basis at x.

III-6.11 ... 111-6.15 6.11

27

H =d H and H = H .

Proof. Assume 6.10. Consider a mapping of a directed system having one element onto x . Since (6 .1 0 ) every nbd of x con­ tains x , this mapping converges to x ; hence xeH implies xeH ; hence HdH . If xeH , then X-H is a nbd of x ; hence (6.10) there is an open set U , such that xeUcX-H . Nov HcX-U , so that (3 .1 a, 6 .2 ) ScX-tJ = H-TJ , hence x«® . Thus ScH , and applying the first relation to H , SdH ; hence H = fl , and 6.11 is proved. Assume 6.11. Let so that xeN . Now xgl-N nbd of x . Now (6 .2 ) U UcX-(X-N) = N . Thus 6.10

N be a nbd of x , then xeX-N^X-N, = X-W , so that U = X-X-N is a is open, and, since X-NdX-N , holds.

A space satisfying these conditions is a T-space. The open sets containing x are open nbds of x . A n alternate ex­ pression of 6.10 is 6.12 N is a nbd of x if and only

We shall restrict the use sets, and the use of U, B, and sets.

if x e U c H

of U,

333 to

where U is open.

V, and W, to open collections of open

The reader may easily show that 6.13 •***. if a family of sets satisfy 6.6, 6.7 and 6.8, and if we call them "open" and use 6.12 to define nbds, then we will obtain a T-space, in which the original family of sets is the f a m i 1y of open s e t s .

A set, its complement 6.14

»***.

H , X-H

|n a

is closed if H = H ; that is (6.2), is open. Prom 6.10, we see that

if

T-space H is a closed set.

We shall often be interested in subcollections of the collection of all open set of a T-space. A subcollection is a basis (for X , for the open sets of X , e t c . ) if every open set is a union of sets belonging to the subcollection. .A sub­ collection is a sub-basis (for X , etc.) if every open set is a union of sets which are themselves each intersections of a finite number of sets of the subcollection. It is easy to see that, 6.15 **•*. |f {U|AI is a basis for X, then, for all x, i U J a s A , x e U j is a nbd basis at x. 6.16 **•*, |f fUa|A1 is a sub-basis for X, then, for all l U J a ^ A , x e t l j is a nbd sub-basis at x.

x,

28

CONVERGENCE

5.17 ****. |f|HJA| is a nbd basis at tion of open nbds of x such that, for then iUb|B1 isa nbdbasis at x,

x, and [UJB] is a c o l l e c ­ each a and some b, ^crN#;

6.18 ***♦. |f{Uc|CI is a sub-basis for X, then the to pology of X is described by the co nv ergence of C-phalanxes. That is, 6.19 x e H < = = ^ f o r

some x("f|C) converging to x, x(r)c=H.

6.20 N is a nbd of x i

-s if x(

|C) converges to x, then

x(DnH^j.

From 4.3 c), 4.3 d) and 6.18 we see that 6.21 ****. The topology of a T-space with a countable sub-basis is determined by the conver ge nc e of simple sequences.

closed.

A space is discrete if every subset is both open and Clearly,

6.22 ****. A discrete space

7.

Continuity.

is a T-space.

We have,

7.1 Theorem. If f is a fv nction defined on the space X and taking values in the space Y, the following are equivalent, 7.2 f(H)c=fTin, for all HczX. 7.3 xfaj-g-^x

implies f (x ( a ) )-£+f (x ).

7.3pp x(6)-~»x, where D - 2 X , implies f (x(6) )-j*f ( x ) • 7.4

If N is a nbd of f(x),

then f*(N)

is a nbd of x.

Proof, 7.2 implies 7.3, for let x(a)-g-*x . Then (3.4), G 1 is cofinal in 6 , x e x ( ; hence (7 .2 ) f (x)cf (x( A» ) )cf (xAf ) ; hence (3.4) f (x(a))— -*f (x) .

If

7 .3 obvious implies 7 .3 pp. 7.3PP implies 7.4, for let N be a nbd of f(x) . If Then (3.6pp), whenever y( 6 )---^f(x) , Nny(A) £ 0 . x ( 6 )---*x , then (7 .3 pp) f(x(&))---►ffx) ; hence Nnf (x( a) ) £ 0 , whence f_1 (N)ox(a) ^ 0 . Hence (3.6pp) f-^N) is a nbd of x . 7.4 implies 7.2, for let f(x)ef(H) . Therefore (3.8) every nbd of x meets H « Now let N be a nbd of f(x) ; then (7.4) f-i(N) is a nbd of x ; hence it meets H ; hence f(f _1 (N) =_N m e e ts f(H) . Hence (3-8) f(x)ef(H) , and this proves f(H)cf(H) . The function f is continuous if it satisfies these conditions. The usual properties of continuous functions can be

III-7-^ ... III-8.6 easily derived from these conditions. ties are 7.5 ****, Every constant

29

Some noteworthy proper­

is continuous.

7.6 ****. a continuous f u n c ti on of a continuous f u n c ti on continuous function. 7.7 **•*. The

identity

is a continuous function

is a

(here Y-X).

Prom 6.4 we easily see that 7.8 ****. 7.9

|f f is contin uo us ,

then

If V is open in Y, then f*(V) is open in X. If Y is a T-space, and if 7.9 holds, then f is continuous*

If f and f"1 are both single-valued and continuous, then they are homeomorphlsms. If f is a homeomorphism, and f(X) = Y ; then X and Y are homeomorphic. Regarded as spaces, any two homeomorphic spaces are abstractly identical. 8.

Separation Axioms. We mention two separation axioms briefly.

8.1 ****.

If X is a T-space,

the following are equivalent;

8.2 Every point of X is closed. 8.3 For each

x, x

Dili |N] a nbd of x ].

B A There is a basis x " e U 4 , then x *=x" .

lUIA} for X such that, if x'eeU

inplies

A T-space satisfying these conditions is a T x-space. A space satisfying 8.5 If F f and F ” are disjoint closed sets, then there are disjoint open sets U 1 and U ” , such that F ’d U * and F Nc = U ” .

is normal. We need the following result later 8.6 * * * * . xsV,

|f {(Ji is a basis for the norial T-space X,

where V is open,

then there are

sets U* and (J**

and

if

in

IUI

• uch t h a t xe sU * where R a e 9 t and each R a e= Then, for some a ’ , x e Vai ; let a contain all a for which V a n Va , £ 0 . If a # oc , the only set of % a = V alXa meeting V at is S(X-Va ,Ua) ; hence, for a ^ a , R a = S(X-Va ,Ua) => V a . . Nov (111-6.7) W = Va,n (fl {Ra |a e a ) is open, and x e V c R . Hence R is a nbd of each of its points, and (ill-6 .4) open. This proves the lemma. 5.2

Lemma.

e a c h U# i s

tf

{VjA}

normal,

is

a sta r-fin ite

then

AJV^UjA}

is

open c o v e r i n g , and

if

normal.

Proof. Since lla is normal, there exists a normal sequence 0V> with XL.< "U* • Now (2.19) V a . Since all these coverings are open (5 .1 ), we see that A { VaoUall I A is a normal sequence. Since A { V a ° H ajl A} < {VaUa|A , the lemma is proved. 5.3

Theor em.

e ac h of

its

A sta r-fin ite associated

covering U

binary coverings

is

nor mal

is

P r o o f . Let U = {TJa |A} ; then is normal, then each is normal.

if

and o n l y

if

normal.

for each

a .

If U

Suppose that each is normal. Let Va = S(Ua,lX) ; then U a n S(X-Va ,U) = 0 , whence V U ca3 ~ U (a) Hence (2.5, 2.3,' 2.4) U ~ A M (9l)IA} ~ A {Va«»U(a)| A}. Now (2.22) U * = {Va (A} is star-finite, so that (5.2) A{Va ° U (a)|A} is normal. Hence U is normal. 5A

The or e m.

A space

(open) co v e rin g

is

is

nor mal

if

and o n l y

if

every s t a r - f i n i t e

normal.

Proof. Normality of the space implies (4.3, 5-3) that every star-finite (open) covering is normal. Since every binary cov­ ering is star-finite, the remainder of the theorem follows from 4.3.

6 . Typical Refinement. 6*1 h e or o reem. a. 6 .1 T The

If U

i s aa ssttaar r- -f fi inni ittee nor n o rmal aal

c o v ec roi vn egr, i n tgh,e nt h ietn

it

has a s t a r - f i n i t e is s t a r - f i n i t e .

and orm I ssttaarr rreeffiinneeameenntt UU11, , ssuucchh tthhaatt UvU* and nnor mal UvU*

P r o o f . Let 'U Then (5*3) each which (4.2, 2.16) the proof of 5.2, refinement of IX

and Va be the same as in the proof of 5 .3 . ix(a; has a normal star refinement , may be taken to be a finite covering. As in we see that = A { Vao U ftl} is a starand that U* is normal. Fix a f, then,

NORMALITY

50

since i . If a ol , then Va n V a . = 0 , so that Ra e V a ° ^ a 4 must be S(X-Va , 11^ ) . Hence the different R fs meeting U a i differ only in the Ra !s for a $ a . Since is finite for each a , it f ol­ lows that U a» meets only a finite number of sets of U ! , and, since U f< U > the theorem follows. 6 .2 ****. E v e r y s t a r - f i n i t e normal c o v e r i n g i s t h e f i r s t member o f a nor mal s e q u e n c e w h i c h i s a s t a r - f i n i t e c o l l e c t i o n of cover ings.

7. Special pseudo-ecarts and pseudo-metrics. A real valued function of two points in X Is a special pseudo-ecart or spe if

1° 2° 3°

f(x,x) = 0 , f(x,y) = f(y,x) ^ 0 . for every e > 0 ,f(x,y) < e and f(y,z) 0 , (y|f(x,y) < e} Is an open set. The next lemma is a trivial generalization of a simple and important lemma due to Prink; the reader is referred to Prink 1937 (p. 134) for the simple proof. 7 . 1 Lemma. o f X, t he n

If f

is

a spl,

and

if

, • • • , x ^ are

any n+2 p o i n t s

In p a r t i c u l a r ,

If we replace 3° by the stronger condition f (x , z ) * f(x,y)+f(y,z) then a function satisfying 1 °, 2°, 3 *, and 4° is a pseudometric.

3*

7 . 2 **♦*. a p s e u d o - m e t r i c i s c o n t i n u o u s i n bot h v a r i a b l e s together. ( T h a t i s , g i v e n x ' t x" and e > 0 , t h e r e a r e open s e t s U x 1 and I T s x" s u c h t h a t , i f x * e U ' and x M e « " , t h e n |f (x f ,x")-f(x*,x**)| and set f(x,y) = 2 ~n if x e S ( y , E a) and x ^ S ( y , IXjjJ • Then f is clearly a spe, and {Un> is a s ­ sociated both with f and with the pseudo-metric connected with f- by 7 .3 . We observe that 7*5

x e 3 ( y , l l n)

imp!ies

r ( x , y ) > 2 “(n+2‘.

The proof of the remainder of the theorem is left to the reader.

8.

Metrlzatlon. A metric is a pseudo-metric satisfying 1* f(x,y) = 0 Is equivalent to x - j ,

and

5° f(xn ,x)— ►() implies x n — ► x . We remark that continuity of f (7 .2 ) implies the converse of 5° . The sphere of radius e (in the particular metric about x is defined by 8.1

the f )

S ( x , e ) = { x 1| f ( x , x 1) < e } .

The spherical covering of radius e is defined by 8.2

lZ (e)={S(x,e)|xeX }.

We clearly have (3*) 8 . 3 **** .

S ( x , e ) c S ( x ,U(e))c= S ( x , 2 e ).

.We easily see that 8.1 ****, A sp6 s a t i s f i e s 5 ° i f c ia t e d sequences, U nf s a t i s f i e s 8.5

For a l l

x,

{ S ( x , U n ) I n & N}

and o n l y

if

i s a nbd b a s i s

one o f

at

x.

its

asso­

52

NORMALITY Nov we have,

8 . 6 Theor em. lent , 8.7

X is

In a T - s p a c e

t he f o l l o w i n g

conditions

are eq u iva ­

metr i z a b l e .

8.8 There e x i s t s a c o u n t a b l e c o l l e c t i o n ings s a t i s f y i n g 8 . S.

| l l n}

8.9 There e x i s t s i ng 8 . 5 .

of c o v e r i n g s

a nor mal

8.10 T he r e e x i s t s and 8 . 5 All 8.11

X is

sequence

a sequence

these c o n d it i o n s a nor mal

{lZn}

are

of

of c o v e r i n g s

implied

nor mal

cover­

satisfy­

s a t is fy in g 3.4

by

space with a countable

basis.

P r o o f . If 8.7 holds, then (7.4) the metric Is associated with a normal sequence, which (8.4) satisfies 8 .5 ; hence 8.9 holds. Conversely, if 8.9 holds, then (7-4) there is an associated pseudo-metric, which (8.4) satisfies 5°; hence 8 .7 holds. If 8.8 holds, then there exist normal coverings so that S8ml * < 33^ 11* , for S8n , 01* and therefore are normal. It is clear that is a normal sequence, and (ill-6 .17) satisfies 8.5; hence 8.9 holds. Clearly 8.9 implies

8 .8 . Now 8.9 clearly implies 8.10. If 8.10 holds, then (3 .3 ) are normal and (ill-6 .1 7 , 3 .5 ) satisfy 8 .5 ; hence 8.9 holds. If 8.11 holds, then (4.7) the countable collection of basic binary coverings satisfies (8 .5 ), so that 8.8 holds. 8.12

Theor em.

Every

(open) co v e rin g

of a m e t r i z a b l e

space

is

norma 1.

P r o o f . Let -X be the metrizable space, and let f be some metric for X . Let U = {Ua |A} be an open covering of X . Now (8.2) for each x , some e > 0 , and some a , x e S(x,e) c: U a . We choose e(x|X) and a(x|X) so that 0 < e(x) < 1 ;

S(x,4e(x)) c: U a(x ) .

Then J8 = {Vx )X} , where Vx = S(x,e(x)), is an open covering of X . We shall prove that $ is normal, then, since we may apply 9-2 to X - U 1 and U .

S(X-U',U) = U",

10. Historical remarks. The notions of *< , , then S(x,lll) = U . If these equivalent conditions are satisfied, then ve say that the uniformity agrees vith the topology. It is clear that if {11} agrees vith the topology of X , then so does every finer uniformity. Not every T-space possess a uniformity agreeing vith the topology. In fact ve have, 3 * 5 Theor em. equ i v a l e n t :

The f o l l o w i n g

c o n d i t i o n s on a T^-space a r e

58

STRUCTS

3.6 I f x f e U f and Uf i s open, valued f u n c t io n h with

then t h e re

3.7

then

If

x 'e lP

and Uf

is

op en ,

is

a continuous

{ X - ,uS

real­

i s a nor mal

cover i ng• 3.8

Some u n i f o r m i t y gX o f X a g r e e s w i t h t h e t o p o l o g y of X.

3.9

The u n i f o r m i t y

3.10

aX a g r e e s w i t h t h e t o p o l o g y o f

The u n i f o r m i t y

fX ag re es wit h th he e t o p o l o g y of

X. X.

P r o o f : We knov (V-9.3) that 3.6 and 3-7 are equivalent, since x ’ is closed, as a point of a Ti-space. Assume 3*7, and let U be a nbd of {X-{x} ,U} ,U} is is normal normal and and x . Then (3-7) be­ longs to fX .Since the star of x in this covering is U , 3.10 holds. 3.10 clearly Implies implies 3.8. 3 .8 .3.3.8 8 implies »9 > sincesince implies3 3.9, (2.13) aX is finer than gX . We need now only prove that 3.9 implies implies 3.7. 3.7. Assume Assume 3-9, 3-9, and and let let x'x' e e U U1 1 . . Then, Then, for for some U e s a X , S ( x !,UQ c: U f . This means that normaland and3.7 3.7 holds. U < { X - { x ’> ,U'} , and so {X- {x1} , U !} isisnormal A Ti-space satisfying these equivalent conditions is Is a completely regular space (Tychonoff). We may derive several consequences from 3.5* We begin with 3.11

Lemma. Leaaa.

Proof.

Every

{X- {x} ,U}

nor mal

T-space

is com pletely r e g u l a r .

is a binary open covering and hence normal.

3.12 R e mark. For Ti-spaces we may regard 3.7 as a natural di­ lution of the requirement of normality (when this requirement is normal.”) is expressed as ’’every binary open covering Is 3 . 1 3 Lemma. I f a T - s3ppa a cce e X satisfies t h e n s o do i t s s u b s p a c e s .

any of 3 . 6

through 3 . 1 0

P r o o f . This is clear (III-5.2, (III- 5 .2 , III-7.3) for 3 .6 , hence (3-5) (3.5) it is true for all. For the remainder of this chapter we will consider only completely regular spaces. Hence we will use ’’space” to mean ’'completely regular space.” 3.14 D e fin it io n . A s t r u c t i s a ( c o m p l e t e l y r e g u l a r ) s p a c e and a u n i f o r m i t y of t h a t s p a c e w h i c h a g r e e s w i t h t h e t o p o l o g y of t he s p a c e .

We refer to the struct, loosely, in the same way that we refer to the uniformity. Thus aX is a struct as well as a

VI-3.14 ... VI-4.3 uniformity. Ve use the space X .

gX ,

hX , ...

59

for general structs over

4. Structs. A basis [sub-basis ] for the uniformity of a struct is a basis [sub-basis] for the struct. A basis consisting of finite coverings is an f-basis. gX

The coverings belonging to the uniformity of a struct are said to be large in gX .

If Y is a subspace of X , and X , then ve may define a struct gY by 4.1

35 i s

large

in gY i f

ft=Ur»Y, wher e U

gX

is a struct

is large

over

in gX.

Ve call gY a substruct ofgX if 4.1 holds. Itis important to notice that if Y is a subspace of X it need not be true that aY is a substruct of aX , or that fY is a substruct of fX . And it may happen that Y is a subspace of X and gY is a struct, but there is no struct hX of vhich gY is a substruct. Ve often deal vith a struct gX and a fixed {11(a)) A} of gX . Under these conditions, ve often S(H,U(a)) to S(H,a) . If gY is a substruct of {II (a)|A> is a basis of gX , then {Yn)X(a)]A} is gY .

basis abbreviate gX , and if a basis of

Ve see that ve have, 4*2 * * * * .

a necessary

c o n v e r g e t o x, s uc h t h a t basis for

is

and s u f f i c i e n t

the e x i s t e n c e

imp 1 i e s

condition

o f an i n t e g e r

x ((5) b

V e may easily recognize 6.2 as a natural generalization of a vell-knovn condition for the convergence of sequences of sets. Since 6.3

**♦*.

2 UI

T is

vas a compact struct, ve have a conpact

struct.

Ve need the folloving result later, 6 .» * * * * .

If

C ( n + I ) => C ( n ) f o r

all

n,

then C ( n | M ) co n v e r g e s

to

U {C(n)|H >.

W e may regard 2 ,AI aa a ring, defining the "sum" of B and C to be (BoC)-(BnC) , and the "product" of B and C to be B C . It is easy to see that these operations are continuous in the topology ve have introduced. Thus 2 ,A1 is a topological r i n g .

Chapter

V III

EXAMPLES 1.

I ntroduction.

2.

Some Hon- n o r ma l

3. 4.

Some S i m p l e S t r u c t s . Some M i l d l y Compl ex S t r u c t s .

Spaces.

1. Introduction. Ve collect here a few examples of interest from a rather general point of view. I.I I n t h i s c h a p t e r we remove t h e r e s t r i c t i o n t h a t 7 i s a f i n i t e s u b s e t of C. We use 7 f o r any s u b s e t o f C, and T f o r any c o l l e c ­ t i o n o f 7 1s .

2. Some non-normal spaces. Ve use a simple modification of an Idea of fiech to exhibit some examples of spaces vhich are com­ pletely regular but not normal. Let C be any uncountable set (e.g., the real numbers), and consider T = 2 ,CI , with the topology discussed in VII- 6 . Let r2 = { 7 1 7 is countable} ; then V2 is a subspace of . Ve now consider the product P ( T 1 ,r2 ) . V e shall write its elements as ordered pairs. Ve consider two subsets of the product, d = i h , y ) \ y ^ T 2} E = {( 0 ,7 )17 €=rz}

,

(c

does belong to

T± !).

I say that these sets are both closed. D is closed, for if (7 («IA) ,7 (cy) A ) ) converges to (7 ’,7") it is clear (VII- 6 .2) that 7 * = 7 ” . E is closed, for the point C is closed in T1 . Let U and V be open sets with D cz U and E cz.V . Let 7 ( l ) e r2 . Let 9 (7 ) = 7 , then 9 (7 )0 ) converges to C , and hence (cp(7 lC) ,7 (1 )) converges to (0 ,7 (1 )) ^ V . Hence there is an 7 (2 ) 1 2 7 (1 ) for which (^(2), 7 (1 )) e V . Ve repeat this argument to obtain a simple sequence (7 (n+l) ,7 (n)) e V , and 7 (n+l) zd 7 (n) . Let 7 ’ = U{7 (n)|N> ; then (VII-6.4) both 7 (n) and 7 (n+l) converge to 7 1 asn -► 00 . Hence (7 (n+l) /y(n)) converges to (7 f, 7 f) as n - > 00 . But ( y ' f y 1) e D c U , hence, for some n , (^(n+l) /-y(n)) e U ;

{y (n)l N} , so that

78

VIII-2 ... VIII-4.2 TJnV ^ 0 .

hence

Thus

Pfl^,^)

Nov (VTI-3, VTI- 6 ) vay, and so is P ( r 4,Ij) ; regular space. So ve have 2.1

P(r

fT )

79

is not normal.

r± = 2 ,CI is a struct in a natural hence the product is a completely

is completely re g u la r

but

not n o r m a l .

If ve regard T4 as a ring, then r2 is a sub-ring, and ve make P( rl,r2 ) a ring in the natural vay. Hence 2.2

****,

There e x i s t

non-normal

topological

rings.

By replacing T (T ,CI vas homeomorphic to 2 ,CI (VIII- 6 )) by, l) the group of the real numbers modulo 1 (the circle group), or 2 ) the group of all real numbers, it may be shovn that 2.3

****.

gical 2.4

There e x i s t

connected,

completable,

non-normal

linear

non-normal

topolo­

groups.

****.

There e x i s t

topological

spaces.

3. Some simple structs. We assume that the topology of the closed interval of real numbers [0,1] is knovn. We shall be interested in tvo of its simpler subspaces. Let X consist of the points l/n, vhere n is a positive integer. Then X is discrete and fully normal; hence every finite covering of X is open and normal. Let Y consist of the points of X and the point 0 . Each open covering of Y contains an open set containing 0 , and hence containing all points l/n for n greater than some n ’ . Suppose that gY is a struct, and that gX is a sub­ struct of gY . Then each large covering of gX contains a set containing all the points l/n for n greater than some n 1 . fX lacks this property; hence fX is a substruct of no gY . 4. Some mildly complex structs. We start vith any uncountable set C , and the family re of its countable subsets (l.l). Zorn's Lemma asserts that there exists a maximal linearly ordered (by zd , of course) subfamily of Te . Let one suc*i be T . We shall need 4*1

If

yne r

f o r e a c h n, t h e n

U{ 7 n | N}

To prove this, let *7 1 = U ^ ^ j N } , and let r . If, for some n , y c z y n , then y T0 - but not Tx-, T^.- but not Hausdorff, Hausdorff but not regular, regular but not completely regular, completely regular but not normal, normal but nor com­ pletely normal, completely normal but n o t perfectly normal, p e r ­ fectly normal but not metrizable, and so on. Are all these categories important? Shall we divide our energies among them all? To me it seems that only a few are importantj this can be verified only after more experience; but I feel that one can now see dimly which are the ones that matter.

84

DISCUSSION

The most general class of object that I would suggest for topological study at present Is what Is defined in I I I -3.16 as a space. We require, for example, of the closure operator only that it commute with union and preserve the empty set. I am not prepared to assert that this is ultimate generality, but rather that this is a natural class of sufficient, generality for present purposes. It may be true that T-spaces form an important class; I am inclined to doubt t h i s . The next important class is that of completely regular spaces, which contains all algebraic objects with an adequate topology. The importance of this class comes from the fact that on them we may erect structs. Since we may build two topologi­ cally invariant structs (which coincide in the compact case) on every completely regular space, it is not unreasonable to say that if a space is completely regular we should usually regard it as a struct. I do not think that normality is important. The intro­ duction (V-3) of the notion of a normal covering should allow many of the important results about normal spaces to be extended to completely regular spaces. And on the other hand, there exist examples (V I I I -2) of various kinds of algebraic objects which fail to be normal. I believe that no algebraic condition (not involving or implying finiteness or countability) can ensure normality. Pull normality remains to be investigated; it may be that it has importance in dimension theory and related topics. Metrizability is of uncertain importance; a large part of its present position may come from its implication of full normality. We have yet to discuss the requirements that a T-space be T0 -, Ti-, Hausdorff, or regular. I allot to these a very minor role. They are convenient if another hypothesis allows one to pass from them to complete regularity (as in the theorem that every Ti-topological group is completely regular). They may perhaps be useful in certain special places. To one who has seen "Hausdorff space" so many, many times this may seem rather harsh treatment; but if he examines a few examples I believe that he will eventually agree with this position. I close this discussion by pointing out the need for a shorter and more suitable term than "completely regular space," and proposing the term "Tychonoff space." 4. No transflnite numbers w a n t e d . I believe that transfinite numbers, particularly ordinals, hav*»a proper place only in descriptive theories, such as: the successive derivatives of a

IX-4 ... IX-5

85

set, the Borel classes of sets and the Baire classes of func­ tions, and some of the less pleasing parts of the theory of d i ­ rected systems. We have succeeded in eliminating all infinite ordinals from the treatment of the subjects dealt vith here. The only transfinite cardinal to make an essential appearance is and ve have already seen that denumerability does play an important and distinctive role. Z o r n ’s Lemma serves to eliminate arguments by transfin­ ite induction. The other stronghold of the transfinite in topology has been the construction of counterexamples. In V I I I -2 and V I I I -4 ve see tvo vays of adapting examples, original­ ly constructed vith transfinite ordinals, to examples construct­ ed vith sets. 5. The subsequence and its generalizations. The stumbling block in the effort to generalize satisfactorily the convergence of sequences vas the subsequence. We may define a subsequence in several vays, and, if ve try to generalize the vrong one, the results may be unfortunate. We may look on a subsequence as a cofinal part of sequence, as a part that goes beyond any given point. If realize that linear order is inessential, then ve are led the ideas of Moore and Smith and to a satisfactory theory convergence.

a ve to of

We may look on the statement ’’every sequence contains a convergent subsequence” as a vay of stating that there are one or more points from vhich the sequence ’’cannot tear itself avay.” If ve adopt this viev, ve are led to cluster points and a satisfactory theory of compactness. W e may look on the statement ’’every sequence contains a convergent subsequence” as being in the form ve vish to general­ ize. If ve do, ve find that it vill not generalize satisfac­ torily. W e may look on a sequence as a "point-set” having the same cardinal number as the ’’point-set" corresponding to the vhole sequence. If ve do, ve are led to the notion of "complete limit-point” and the complications of infinite cardinal arith­ metic. This leads to a complicated and non-perspicuous theory of compactness. This last point of viev vas encouraged by tvo details. First, the a n al y s t ’s habit of saying ’’for infinitely many n," vhen he meant ’’for a subsequence of n . ” As long as one is solely concerned vith sequences, this is a simple and suitable method of expression. From a general point of viev, hovever, this expression emphasizes the vrong property. Second, in talk­ ing about a sequence, in topology but not in analysis, there vas

86

DISCUSSION

a tendency to neglect the ordering; to regard a sequence as a ’’point-set” rather than as a function defined on the positive integers. This attitude is exemplified by the capeful (some­ times.1) distinction betveen a ’’Punktfolge" and a Zahlfolge.” It is my firm conviction that this last point of viev is not a good one, as judged by its results; for I feel that cardinal arithmetic of the complicated sort (’’regular alephs,” ’’accessible alephs,” etc.) should be kept as far from general topology as possible. This is the ordinal part of the theory of cardinal numbers, and is essentially descriptive. It is not the task of general topology to describe objects in terms of ordinal numbers. 6 . Phalanxes vs. Fi l t e r s . The notion of a filter, introduced by H. Cartan in 1937* has been used by N. Bourbaki and his col­ laborators as a generalization of a sequence. That is, they consider the ’’convergence” of filters. Given a sequence, they consider the filter of all sets vhich contain the image of a ’’residue” of the integers. The filter converges to a point if it contains every nbd of the point. Thus for filters derived from sequences ve have the classical convergence condition. This notion has the disadvantage that, in the case of sequential convergence, it concerns itself vith the unintuitive family of sets, rather than the rather intuitive sequence. Thus, to obtain generality, ve must abandon the intuitively satisfactory treatment of the sequential case. It may seem u n ­ fortunate, to one interested in generalities, that the sequence has a special place in topology. It seems, hovever, to be true. (We may alvays replace the sequence by a phalanx on a countable base, but in a fev places this seems artificial). Phalanxes, on the other hand, form a part of a theory of convergence (of functions on directed systems) vhich includes sequences. We obtain generality vithout discarding the intui­ tive treatment of special cases. In closing, it should be noted that the idea of an ultraphalanx stems from the idea of an ultrafilter, and that they are essentially equivalent. The equivalence is betveen a single ultrafilter and a class of ultraphalanxes. Ultrafilters have the advantage of uniqueness. Ultraphalanxes have the a d ­ vantages of occasional simplicity and inclusion in a simple theory of convergence.

Bi b lio g ro p h y Alexandroff and Urysohn 1933P. Alexandroff and P. Urysohn, "Sur les espaces topologiques compacts.” Acad. Scl. Cracovle Bull. Int. (also known as "Bull. Int. Acad. Polonaise Scl. et Lett.) 1923, pp. 5-8 (1924). Banach 1932. S. Banach, "Theorie des operations lineaires." Matematyczne 1, (Warsaw).

Monografje

Birkhoff 1937. Garrett Birkhoff, "Moore-Smith convergence in general topolo­ gy." Annals of Math. 38 , pp. 39-56. Birkhoff 1939. Garrett Birkhoff, "An ergodic theorem for general semi-groups" Proc. Nat. Acad. Sci. 25, pp. 625 -627 . Cartan 1937. H. Cartan, "Theorie des flitres" and "Filtres et ultraflitres." Comptes Rendus (Paris) 205, PP. 595-598 and 777779. Cohen 1937. L. W. Cohen, "Uniformity properties in a topological space satisfying the first denumerability postulate." Duke Math. J. 3, PP. 610-615. Cohen 1939. L. W. Cohen, "On imbedding a space in a complete space." Duke Math. J. 5, pp. 174-183. Dieudonne 1939. J. Dieudonne, "Un example d'espace normal non susceptible d ’une structure uniforme d !espace complet." Comptes Rendus (Paris) 209, pp. 145-147. Frechet 1906. M. Frechet, "Sur quelques points du calcul fonctionnel." Rendiconti Palermo 22, pp. 1-74. Frechet 1921. M. Frechet, "Sur les ensembles abstraits." 38, pp. 341-385.

Ann.

Ecole Norm.

Freceht 1928. M. Frechet, "Les espaces abstraits." Villars.

87

Paris, Gauthier-

88

BIBLIOGRAPHY

Prink 1937. A. H. Prink, "Distance-functions and the metrization problem." Bull. Amer. Math. Soc. 43, pp. 133-142. Graves 1937. L. M. Graves, "On the completing of a Hausdorff space." Annals of Math. 38 , pp. 6l-64. Moore 1915. E. H. Moore, "Definition of limit in general integral analysis." Proc. Nat. Acad. Sci. (Washington) 1, pp. 628-632. Moore and Smith 1922. E. H. Moore and H. L. Smith, "A general theory of limits." Amer. Jour, of Math. 44, pp. 102-121. Weil 1936. A. Weil, "Les recouvrements des espaces topologiques: espaces complets, espaces bicompacts." Comptes Rendus (Paris) 202, pp. 1002-1005. Weil 1937. A. Weil, "Sur les espaces a structure uniforme et sur la topologie generale." Actualites Sci. Indus. 551, Paris, Hermann et Cie.

INDEX f a c t o r , i nessent i al f ac t o r , 7 f i n e r than (of coveri ngs ) , 57;

ath coordinate, pr oj ect i on on the

coordinate, 71

( of topol ogi es ) , 24 f i n i t e , f i n i t e character, 7; coveri ng, 46; r e s t r i c t i o n , 7; set, 3

associated sequence of cover­ ings, 51 base o f a stack, 13 b as i s , 27; f or a s tr u c t , 59; f or a uniformity, 56

f u l l y normal, 53

binary covering, 31 bounds: upper, lower, 4

homeomorphic, homeomorphism, 29

car di nal number, 9

i nessent i al f ac t or , 7

Cauchy mapping, 62 Cauchy phalanxes, pa i r ( of

infImum, 4 i nf l at e d phalanx, 17 i nt e r se ct i on, 2 irreflexive, 3 isomorphic,' 3 it er at ed s t a r , 44

h ype r s l i c e , 71

phalanxes), 63 cl osed set , 27 c l u s t e r point, 34 c o f i n a l , 10; cof i nal types, c o f i n a l l y s i m i l a r , II

12

l arge coveri ng, 59 l a r g e l y compact, 60 l i n e a r ordered system, 3 lower bound, 4

compact, 37; compact s tr u c t , 60 complete s tr u c t , 64 complete t r e l l i s , 5 completely r egul ar, 58 completion of a s t r u c t , 64 continuous, 28; uniformly con t inuous, 61 continuous image, 37 coveri ng, 31; associated sequence

mapping (Cauchy), 62 maximal, 7 meets, 2 meshed base of a phalanx, |7 metric, 51 nbd: b asi s, sub- basi s, 24; open, 27 normal coveri ng, normal sequence of coveri ngs, 46

of coveri ngs, 51; basi c binary, 48; binary, 31; equi val ent, 44; f i n e r than, 57; f i n i t e , 46; i nt e r se ct i on of, 43; l arge, 59;

normal space, 29

normal, normal sequence of, 46; refinement, 43; s t a r - f i n i t e , s t a r - f i n i t e c o l l e c t i o n of, 46;

decided about, 32 decides about, against, f or , 32

open nbds, 27 open set, 26 ordered system, I r r e f l e x i v e , isomor­ phi c, l i n e a r , properl y ordered, r e f l e x i v e , symmetric, t r i v i a l l y ordered, vacuously ordered, 3

d i s c r e t e , 28 disjoint, 2

par al l e l ot ope , 75 p a r t i t i o n , 31

union of, 43

equi val ent coveri ngs, 44

89

INDEX

90

phalanx, 17; Cauchy, Cauchy p a i r of, 63; base of a, i n f l at e d , meshed, 17; of rank n, 65 pointwise product, 72 product, 6; the product, 7; topo­ l ogi cal or pointwi se, 72 pr oj ect i on on the a**1 coordi nate, 71 properl y ordered, 3 pseudo-metric, 50

subspace, 26 substr uct, 59 subsystem, 3 superspace, 26 supremum, 4 symmetric, 3 system of subsets, 3 Tj - space, 29 t opol ogical product, 72 topologyi 24 topophalanx, 33 transitive, 2

refinement, 43 ref lex iv e, 3 r e s i d u al , 10 r e s t r i c t i n g , 45 space, 24; d i s c r e t e , 28; normal, 29 speci al pseudo-ecart, or spe, 50 stack, base of a stack, 13 s t a r , iterated s t a r , 44; s t a r - f i n i t e covering, s t a r - f i n i t e c o l l e c t i o n of coveri ngs, 46; star- ref i nement, 45 s t r u c t , 58; basi s f or a, 59; com­ pact, l ar g e l y compact, 60; com­ pl ete, completion of a, 64 sub-, 2 s ub-basi s, 27; f or a uniformity, 56 subcovering, 31 subphalanx, 35

t r e l l i s , complete t r e l l i s , 5 t r i v i a l l y ordered, 3 u l t i mat el y, 32 ul traphal anx, 32 uniformity, 55; basi s f or a, induces a, sub- basi s f or a, 56 uniforml y continuous, 61 unimorphism, 61 union, 2; of coveri ngs, 43 upper bound, 4 vacuously ordered, 3 Zorn, 7

SYMBOLS

2*,3 ( r , a cl c ) , P {a c| c l ,7

It >» 12 X('ylc),

17