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8526149

Burdick, Bruce Stanley

LOCAL COMPACTNESS AND THE COFINE UNIFORMITY WITH APPLICATIONS TO HYPERSPACES

The Ohio State University Ph.D.

University Microfilms International300 N. Zeeb Road, Ann Arbor, Ml 48106

Copyright 1985 by Burdick, Bruce Stanley All Rights Reserved

LOCAL COMPACTNESS AND THE COFINE UNIFORMITY

WITH APPLICATIONS TO HYPERSPACES

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the Graduate

School of The Ohio State University

By

Bruce Stanley Burdick, B.S., M.S.

The Ohio State University

1985

Reading Committee: Approved By

Philip Huneke

Francis Carroll Adviser Department of Mathematics Henry Glover Copyright by

Bruce Stanley Burdick

1985 Dedicated to the memory of Dr. Norman Levine, my advisor from 1980 to 1983, whose teaching style and scrupulous attention to detail will be an inspiration to me through out my career. ACKNOWLEDGMENTS

I wish first of all to recognize ray debt to my parents, who from early on encouraged ray curiosity and my appetite for the written word. There have also been many other people, both in the academic community and in the various other communities in which I have been involved, who through their friendship and encouragement have helped see this project through to its completion. Since a complete listing of these persons here would be impossible,

I assure each of my friends that their contributions, regardless of their irrelevance to the subject matter, is remembered and appreciated.

My debt to Dr. Norman Levine is very great. It was he who suggested that I work with uniform spaces; he was the one to whom I first brought each of the results now included in this dissertation and he patiently reviewed each proof, catching my mistakes and suggesting new approaches.

I wish to express my heartfelt gratitude to

Dr. Philip Huneke, who took over as my advisor in 1984, who gave of his time to let me present to him the results

I had found, and who made many helpful suggestions. I

iii would like to thank also the other members of my reading committee, Dr. Francis Carroll and Dr. Henry Glover, for

their patience and understanding. VITA

June 20, 1956 ...... Born - Middletown, Connecticut

1978 ...... B.S., Heidelberg College,

Tiffin, Ohio

1978-1985 ...... Teaching Associate, The Ohio

State University, Columbus,

Ohio 1984 ...... Teacher, Punahou School,

Honolulu, Hawaii

PUBLICATIONS

"From Time to Time." Analog Science Fiction / Science

Fact, October 1983, pp. 114-122. Reprinted in From Mind to Mind: Tales of Communication from Analog,

Stanley Schmidt, ed. New York: David Publications,

1984. "Q.E.D." Analog Science Fiction / Science Fact, December

1984, pp. 96-112. TABLE OF CONTENTS

DEDICATION...... ii

ACKNOWLEDGMENTS...... i ii

VITA ...... v FIGURE ...... vii

INTRODUCTION ...... 1

NOTATION, TERMINOLOGY, AND BACKGROUND ...... 5

CHAPTER ONE LOCAL COMPACTNESS AND THE COFINE UNIFORMITY ...... 21

CHAPTER TWO PROPERTIES OF THE COFINE UNIFORMITY ...... 38

CHAPTER THREE THE HYPERSPACE UNIFORMITY ...... 47

CHAPTER FOUR LOCAL COMPACTNESS OF THE HYPERSPACE...... 56

CHAPTER FIVE CHARACTERIZATION OF HYPERSPACES ...... 73 LIST OF REFERENCES ...... 85

vi FIGURE

0

Page

Figure 1. A Commutative Diagram for Adjointness of m ...... 46

vii INTRODUCTION

The structure known as a uniform space provides the

theoretical mathematician with an entity intermediate between metric spaces and topological spaces. Iii the con

text of uniform spaces the properties of completeness,

total boundedness and uniform continuity can be discussed properties which are usually associated with metric space

Each metric on a set generates a uniformity, each unifor­ mity generates a topology, and there is a certain consistency here in that the topology generated by the uniformity generated by the metric is the same as the

topology generated directly by the metric. On the other hand, there may be several metrics generating the same uniformity or several uniformities generating the same

topology.

The collection of uniformities on a set forms a lat­

tice under the inclusion ordering. So there is always both a largest and a smallest uniformity for a given set.

The subcollection of uniformities generating a given

topology always has a largest element but does not always have a smallest element. As shown by Samuel [14] and

Shirota [16] there is such a smallest uniformity if and only if the given topology is that of a locally compact

space.

Some topologies on a set are compatible with only one uniformity. For example a topology which makes the

set compact is generated by a unique uniformity. But

Dieudonne [4] gave an example of a non-compact space which admits a unique uniformity. So the question arose as to what topological property is equivalent to the space being generated by a unique uniformity. Different answers to

this were given by Doss [5], Newns [13], and Gal [6].

It turns out that this problem of characterizing spaces with unique uniformities is very closely related to the problem of characterizing the smallest uniformity for a space. In Chapter 1 we will give a new proof of the

Samuel-Shirota theorem, and we will give nine statements that are equivalent to saying that a uniformity is smallest for its topology. Notable among these is that if a uniformity is minimal among the uniformities generating a given topology then it is actually the smallest of those uniformities. Many of these statements are similar to or even the same as known properties of spaces with unique uniformities. This then generalizes some of the work on unique uniformities since a unique uniformity is auto­ matically •smallest for its topology. Based on what we have found out about smallest uniformities we will give new proofs of some of the characterizations of spaces with unique uniformities using the simple idea that a uniformity is unique if and only if it is both smallest and largest for its topology. Warren

[19] has used the Tq identification to show that the results of Doss, Newns, and Gal are true even if one does not assume that the spaces are T 2 . Since we will not assume the T 2 property in Chapter 1 we will be proving some of Warren's claims without using Tg-identifications.

In Chapter 2 we will see whether the property of being a smallest uniformity is preserved by some of the construc­ tions of topology, namely, uniformly continuous maps, subspaces, product spaces, sum spaces, and unions. We end the chapter with a discussion of functorial properties.

There is a construction in topology called the hyper­ space, whose points are the non-empty closed sets of a given base space. Hausdorff [7] puts a metric on this space, Vietoris [18] worked with a hyperspace topology, and

Bourbaki [l] defined a hyperspace uniformity. Many properties of the Bourbaki construction were worked out by

Michael [12] and Caulfield [3]. In Chapter 3 we will give an introduction to hyperspaces, proving some properties we will need later. The question of when a hyperspace is locally compact is the theme of Chapter 4. Along the way we will show that a hyperspace uniformity is smallest for its topology if and only if it is compact. Since the union of two non-empty closed sets is a non-empty closed set, the union may be regarded as a binary operation on the points of the hyperspace. In this way, hyperspaces may be viewed as algebraic structures which carry a uniformity. In Chapter 5 we will give necessary and sufficient conditions that such an algebraic structure is unimorphic to a hyperspace. We will wrap up the chapter with some simple properties of such structures. NOTATION, TERMINOLOGY, AND BACKGROUND

This section is intended as a broad survey of the assumptions and conventions used in the text. More infor­ mation can be found in Kelly [10], Willard [20], and

Bourbaki [1 & 2].

Relations. Given a set X, any set R C X x X is called a relation on X. If R is a relation and A C X then

R[A], which is read "R sectioned at A," is the set

{y e X| 3 x e A with (x,y) e R}. If R and S are relations on X then R O S = {(x,z)| By e X with

(x,y) e S and (y,z) e R}. R O S, read "R compose S," is then a new relation. If R is a relation we define

— 1 R = {(x,y) | (y,x) e R}. A number of simple properties of relations will be used without being mentioned; for example R[S[A]] = R O S[A], and the fact that R[A] f] B = tf if and only if A PI R-^[B] - Gf. If x e X then R[{x}] will be abbreviated by R[x].

The diagonal of X, denoted A^, is the relation

{(x,x) | x e X}. Note that A^ O R = R O A^ = R. A rela-

tion R is symmetric if R = R , reflexive if A ^ C R, and transitive if R O R CZ R, and if R is all three then

R is an equivalence relation. An equivalence relation R 6

takes the form U { x a x X a|a e L} where the X a's are called the equivalence classes of R.

Uniformities. A non-empty set U of relations on X is called a uniformity for X if it satisfies the following properties:

i. C ^ ^or ea°k U e 1/ . ii. If U e u then e u .

iii. If V 3 U e u then V e u.

iv. If U,V e a then U fl V s u.

v. If U e u then there is a V e u with V O V C U.

When showing that a given set u is a uniformity it is usually the fifth property which is the hardest to verify. There will be cases in the text where we will say

"the first four properties of a uniformity are trivially satisfied, and it remains to show that the fifth property holds."

Uniform Spaces. If u is a uniformity for X then (X,t/) is called a uniform space. Each uniform space (X,E/) has an associated topological space (X,r(£/)) where r(u), the topology generated by u , is {0 C X| V x e 0, 3U e u with

U[x] CO}. Because of this, topological properties may be attributed to a uniform space, e.g., (X,i/) may be said to be compact, locally compact, etc. For x e X, the neighborhood system of x is exactly the set (u[x]| U e t/}.

f 7

If i/^, u2 are uniformities for X with C then

T ( t/1) C T(U2).

Discrete and Indiscrete. On a set X, {X x X} is called the indiscrete uniformity and (I) C X x X| U} is called the discrete uniformity. The indiscrete and dis­ crete uniformities generate the indiscrete and discrete topologies, respectively.

Basis and Subbasis. A set of relations a is a basis for a uniformity u on X whenever £/={UCXxX| 3 V £ b such that V C U}. This is equivalent to saying a C U and v c {U C X x X| 3 V e a with V C U} . a is a sub­ basis for u if the set of finite intersections of members of a is a basis for u .

Pseudometric Spaces. Given a pseudometric d: X x X -*■ R and a number e > 0, let £ be the relation {(x,y) e XxX| d(x,y) < e } . Then {V, | e > 0} is a basis for a unifor- CL y £ mity t/^ on X. The topology generated by d is the same as

Open and Closed Relations. For a topological space (X,r), a relation R is called an open (closed) relation if it is open (closed) as a subset of X x X with the product topology. If x e X and R is a closed relation, then

R[x] is a closed set. If A C X and R is an open 8 relation then R[A] is an open set.

If (X,u) is a uniform space then the set of closed

symmetric members of u forms a basis for u, and the set of open symmetric members of u forms a basis for u •

Interior and Closure. In a topological space (X,r) we denote the closure of a set A C X by cA, and the interior of A by Int(A).

In a uniform space (X, u) we have cA = H {U[A] | U £ t/}-

For x e X it is true that H {U[x]| U £ £/} = (flE/)[x] but it is not true in general that cA = (fit/) [A]. We have, though, the following:

Lemma 0.1. If (X,u) is a uniform space and A C X is compact, then cA = (fli/)[A]. Proof. Suppose x e cA.

Then for any U e u we have U[x] fl A ^ 0. Since A is compact, the intersection of the sets V[x] fl A, for closed symmetric V e u, must contain a point x*. Then for any U e u there is a closed symmetric Vet; with

V C U, and we have (x,x*) e V so (x*,x) e V and so

(x*,x) e U. Thus (x*,x) e Ht/j and so x £ ( (li/)[x*] C

(flu) [A]. This shows that cA C (fit/) [A]. Conversely,

(nif)[A] C n {u[A] l U e a] = cA.

Separation Axioms. We will not assume that any arbitrary space under discussion is Hausdorff (T2 ) unless that property is specifically stated. In a uniform space (X,t/) the topology r (u) is T 2 if and only if fit; = Ax . In this work the properties of regularity, complete

regularity, and normality will not automatically include

the Hausdorff property.

Uniformizability. A topological space (X,r) is called uniformizable if there is a uniformity u - for X with t = t (u ). A space is uniformizable if and only if it is completely regular. The most important consequence of

this for us is that uniform spaces are always regular.

Local Compactness. The statement that x e X is a point of local compactness of (X,r) has three meanings:

a. x has a compact neighborhood.

b. x has an open neighborhood with compact closure.

c. Given a neighborhood N of x there is an open set 0 with x e 0, cO C N, and cO compact.

These definitions are not equivalent in general.

However they are equivalent when the space is assumed to be regular. Since all the spaces under discussion here will be regular, we will use these definitions inter­ changeably. A space (X,£/) is uniformly locally compact if there is a U e U such that for any x e X, U[x] is compact.

If (X,tf) is uniformly locally compact then there is a

V e u such that if A C X is compact, so is c(V[A]). 10

Compact Sets. If (X,r) is locally compact and regular, and

A C 0 C X with A compact and 0 open, then there is

an open set O' with A C O', cO' Cl 0, and cO' compact.

Equivalently, if A C 0 e t with A compact, there is a

compact set B C 0 such that A C Int(B).

If (X,tf) is a uniform space, and A C 0 C X with

A compact and 0 open, then there is a U e u with

U[A] C 0. Equivalently, if A is compact and B is closed, with A fl B = 0, then there is a U e V with u [a] n b =

As a consequence, if A is compact and A C 0 e t(u)

then cA C 0 (since cA Cl U[A] CO). So if C is an open cover of A then c covers cA (since A C Lie and

U c is open, so cA CU c ). Finally we have:

Lemma 0.2. In a uniform space (X,tf) if A C X is compact then so is cA. Proof. If c is an open cover of cA there is a finite subset of c which covers A and which is therefore a finite subcover, covering cA.

Neighborhoods of the Diagonal. In a uniform space (X,t/) each U £ u is a neighborhood of the diagonal with respect

to the product topology on X x X, i.e., A^ C Int(U).

It is not in general true that any neighborhood of the diagonal is a member of £/, nor need it be true that the collection of neighborhoods of the diagonal is a uniformity. 11 Compact Spaces. If (X,r) is a compact uniformizable

space then there is a unique uniformity u for X such

that T = r ( u). Furthermore, this u is exactly the neighborhoods of the diagonal, i.e., u = {u C XxX)

C Int(U) } .

Uniform Continuity. If we are given a map £: X -*■ Y let

the map fxf:XxX-*YxY be defined such that for x^,X2 e X we have, f x f(x^,X2 ) = (f(x^),f(X2 ))• If (X, u) and (Y, v) are uniform spaces we say that f : X Y is uniformly continuous whenever each V e v satisfies

(f x f) [V] e u . Some equivalent definitions are:

1. For each V s v there is a U £ u with

U C (f x f)~1[V].

2. For each V £ v there is a U e u such that at each x e X we have U[x] C f **^[V[ f (x) ] ] .

When (X,u), (Y,v) are uniform spaces and f: X Y is uniformly continuous we will write f: (X, u) ■* (Y,v).

If (X, r) , (Y, s) are topological spaces and f: X Y is continuous we will write f: (X, r) (Y,s).

Whenever f : ( X , u) ■+ (Y,v) then f: (X,t(u)) ■+

(Y ,r(v )). On the other hand if (X, u) is compact and f; (X, r( u) ) (Y, r( v) ) then f: (X, u) - (Y,v).

Uniformly open maps. If (X, u) and (Y, v) are uniform spaces and f: X -► Y, we say f is uniformly open if for 12

each U e u there is a V e v such that at each x e X we have V[f(x)] C f[U[x]]. If f is uniformly open and onto then the following property holds:

UO*: For each U e u there is a V e v with V C (f x f)[U]. (if f is one-to-one and U0“ holds

then f is uniformly open.)

Unimorphism. A bisection f: (X,u) *+• (Y,v) which is uniformly open is called a unimorphism of (X,t/) and

(Y,v). Equivalently, a unimorphism is a uniformly con­

tinuous bijection for which UO* holds. Any unimorphism is a homeomorphism.

Subspaces. If (X,i/) is a uniform space and A C X then

the subspace uniformity on A, denoted i-s the set {(A x A) C U| U e u }. We will use the abbreviation

(Y,v) C (X,t/) to mean Y C X and v =

In a topological space (X,r) we will denote the restriction of t to A C X by r|and we will take

(Y,s) C (X,r) to mean Y C X and s = t |y. It is always true that = r(u|^).

Weak Uniformity. Given a map f: X Y and a uniformity

v for Y, the set V = {U C XxX| 3 V e v with

(f x f)"1[V] C U) is a uniformity for X called the weak uniformity for f and V. We have f: (X,u) ■+■ (Y,v); 13 in fact u is the smallest uniformity which makes f uniformly continuous.

Supremum. If {^a| a e l} is a collection of uniformities for X then its union is a subbasis for a uniformity u on X. We denote this u by sup{^a | a e L}. It is always true that supi^u^) | ot e l} = r(sup{ua | a e L}).

Product Spaces. If ((xa}Ua)| a E L} is a collection of uniform spaces, let X = x{Xa | a e L} , i.e., ’X is the

Cartesian product of the Xa 's, and let Pa be the projec­ tion of X onto Xa . Let £/ * be the weak uniformity on

X for Pa and ^a. Then u = sup{c/a*l a e L} is called the product uniformity on X and we abbreviate this state­ ment by (X,tf) = x{(Xa ,ff0)| a e L} . If L = {1,2} we write (X,t/) = (X1 ,^1) x (X2 ,u2).

If (X, v) = x{ (Xa, U0L) | a e l} then (X,r(t/)) = x{(Xar

Sum Spaces. If (X^,^ ),(X2^2) are uniform spaces we write (X,y) = (X^,^) + (X2 ,U2 ) to mear> that X^ n X 2 =* 0 , X = U x2, and t/ = {U C X x X| U n (X^ e and U n (X2x X2) e u }. We have (Xx , , (X2 , u2) C (X,^). Since (X^x X1) U

(X2 x X2) e u, then X^ and X 2 are both clopen sets in X. 14

Normal Separability. In a space (X,r) two sets A,B C X are called normally separable if there is a continuous real­ valued function f:X -► R such that f[A] C {0} and f[B] C

{li­ lt is a useful result in uniform space theory that any uniformity is generated by a certain family of pseudomet­ rics; more specifically, if (X,£/) is a uniform space and

U e u then there is a pseudometric d: X x X ■+ R such that

C u and ^ C U. We assume this fact to prove the following:

Lemma 0.3. Let (X,i/) be an arbitrary space and let (R,w) be the reals with the usual metric uniformity. If A,B C X, U e [/, and U[A] n B = 0, then there is a map f: (X,t/) -*■ (R,w) such that f[A] C {o} and f[B] C {1}- Proof. Take a pseudometric d with u d C u and ^ C U.

Without loss of generality we may assume A f 0 , so the equation g(x ) = d(x,A) defines a map g: X -*■ R. When­ ever d(x,y) < e we have |g(x) - g(y)| = |d(x,A) - d(y,A)|S d(x,y) < e and so Vd ^e C (g x g)”1 [Vd

{0} and g[B] C [1,“ ). So define f: (X,t/) •> (R,w) by f(x) = min{l,g(x)}. Then f[A] = {0} and f[B] C {l}.

Nets. If (D,S) is a directed set then a map S:D -*• X is called a net in X. For two directed sets D' and D, a map 15 f: D' D is called a cofinal map if for any d e D there is a d ^ 1 e D' for which 62' £ d ^ 1 implies f(d2 *) £ d.

S ' : D 1 -*■ X is a subnet of S: D -*■ X if there is a cofinal map f such that S' = S O f.

In a uniform space (X,u) a net S: D X is a Cauchy net if given U e r/, there is a d e D such that d^,d2 £ d implies (Sd^»Sd 2 ^ £ U *

For d e D define S[d,°°) = {S^t | d 1 £ d}-. • The set of cluster points of S will be denoted cluster(S). Note that cluster(S) = (~1 {cS[d,°°) | |d e D} . The net S converges to x in (X,tf) if and only if for any U e U there is a d e D with S[d,“>) C U[x]. Any convergent net is Cauchy, and any cluster point of a Cauchy net is a limit point of the net.

Order Notation. In Chapter 5 we will be working with a directed set (D,S) and a partial order (P,£) in the same paragraph. We will exclusively use £ for directed set orderings and ^ for other poset orderings.

Filters. In a space (X,t/) a filter F on X is Cauchy if given U e U there is an F e F with F x F C U. The set of cluster points of F, denoted cluster(F), is n {cF| F e F}. Given two filters F^ and F2 on X, the set F^ H F2 is a filter on X and cluster(F^ n F2 ) = cluster(F^) u cluster(F2 ). A filter F converges to x in (X,£/) if {U[x]| U e v ] C F. 16

Total Boundedness. A space (X,t/)' is totally bounded if for each U e U there is a finite set F C X with

X = U[F]. A partial list of equivalent definitions is:

1. Every net S: D •+ X has a Cauchy subnet.

2. Every sequence S : N -*■ X has a Cauchy subnet.

3. Every ultrafilter on X is Cauchy.

If (X,i/) is totally bounded and is dense in (Y,v), then (Y,v) is totally bounded.

Completeness. The space (X,tf) is called complete if every

Cauchy filter on X converges in X, or equivalently, if every Cauchy net in X converges in X.

(X,tf) is compact if and only if it is complete and totally bounded. Any uniformly locally compact space is complete.

Completion. A completion of a space (X,tf) is a map f: (XjU) -*■ (Y, V) such that f is a unimorphism of X and f[X], (Y,v) is complete, and Y = cf[X].

If f: (X,tf) -*• (Y,v) is a completion and g: (X , tr)

(R,w) is a real-valued function, there is always an extension g: (Y,v) (R,rO such that g = g O f.

Every uniform space admits a completion. The standard proof of this is in Kelly [10], and another proof using hyperspaces can be found in Caulfield [3]. Assuming this fact we have: 17 Lemma 0.4. Given a space (X,u) there is a completion

f: (X,£/) ■+ (Y,v) with the following separation property:

if y^ £ ^2 are points of Y which are not both in f[X]

then there is a V e v with V[y^] n V[y2 ] = 0.

Proof. Let g: (X,t/) -► (Y*, v*) be a completion. n V* is an equivalence relation on Y and the equivalence

classes {^a l a e L} are just the closures of singletons

in Y (see the discussion preceding Lemma 0.1). We wish

to delete points of Y* in the following way: if Ya n g[X] £ 0 then delete Y^g[X] , and if Yq n g[X] = 0

then choose one vpoint jy a e Y a and delete Y a \{y 1J aJ }. Let Y be the set of points not deleted and let V = v*|y.

We did not delete any points of g[X], so g[X] C Y.

Let f: (X,t/) •+ (Y,v) be given by g(x) = f(x). Then f

is a unimorphism onto its image because g was, and f[X]

is dense in Y because g[X] was dense in Y*. If a net

converges to a point y* e Y* then it also converges to

any point of c{y*}= (nv*)[y*]. Since Y contains at

least one point of each (nv*)[y*] we, have that any net

into Y convergent in Y* is also convergent in Y.

So (Y,v) is complete, and this means f: (X,t/) -► (Y,v)

is a completion. (Y,v) satisfies the separation property. For if

x e f[X] and y e Y \ f[X] then y £ (nv*)[x], i.e.,

(x,y) £ D v*. So take V* e v* with (x,y) £ V*. _ 1 V* n (Y x Y) e v so we may take a V e v with V O V C V*. (x,y) i V -10 V so V[y] n V[x] = 0. On the other hand, if e ^ \ then ^ ^ ^ 2 ^ ^ n v* since ( nv*)[yi] contains only one point of Y, i.e., (nv*)[y^] n Y = {y^}. Then as above we can find a V e v with

V[yi] n V[y2] = 0.

Tq -identification. In a space (X,r) let R be the equivalence relation {(x,y) e X x X|c{x} = c{y}}. For x s X,

R[x] is the equivalence class of x. Let P: X -+■ X'* be the projection of X onto X*'% the set on equivalence classes of R. Then P(x) = R[x]. Let r* be the quotient topology on X*. Then (X*,r*) is the T^-identification of (X,r). In a uniform space (X,tf) the relation R is just nu. Let u* = {U* C X* x X*j (P x P r H u * ] e £/}. Then

(X'v,tf*) is a uniform space and . Since

(X*,tf*) is regular and Tq it is T2 . P: (X,u) (X*, u *) and u is weak for P and u * .

Proper Maps. For spaces (X,r),(Y,s) which are locally compact but not necessarily T2 , the appropriate definition of a proper map is the following: f: X Y is proper if _ d whenever A C Y is closed and compact then f [A] is compact. For a discussion of proper maps in the context of

T 2 spaces see Bourbaki [1, Chap. 1, sec. 10, part 9]. 19

Adjoint Functors. Given a category C and two of its objects X and Y, let C(X,Y) be the set of morphisms from X to Y. Then we can regard C( , ) as a functor from C x C to S (the category of sets), covariant in the second variable and contravariant in the first.

Extending this idea, if F^: -*■ C 2 and F2 : C 2 -*■ are covariant functors then ^(F.^ ^ ^ are functors of the form x C2 S. The definition of adjoint functors, due to Kan [9], is: if there is a natural isomorphism y: ^(^iCX^Y) ■+ ^(X, ^ then F^ is left-adjoint to F2 and F2 is right- adjoint to F^. The set of maps {r^ yl ^ z ^ 1 } Y e C 2 } is called the adjunction.

V Hyperspaces. For a space (X,u) let 2 be the set of non-empty closed subsets of X. We will use Script 12 y letters (A, B, C, S, U) for subsets of 2 and relations y on 2 , except in those cases where the notation has been set by Michael [12], e.g., c(X), and H(U).

Proper Partial Orders. A partial order (P,s) is proper if whenever x £ y and y £ x then x = y. A directed set need not be proper.

Groupoids. If *: X x X X is a binary operation on

X, assigning to a pair (x^,X2 ) the point x^ * X2 ,

t 20 then for sets A,B C X let A * B = { x^ * X2 | x^ e A and e B }. CHAPTER 1

LOCAL COMPACTNESS AND THE COFINE UNIFORMITY

One of the theorems of uniform space theory is that

the supremum behaves well with respect to taking the

generated topology, i.e., r(sup{tfa | a e L}) = sup {r( r/^) |

a e L }. As a consequence, given a uniformizable space

(X,r) the uniformity f ( t ) = sup{u| t(u ) = r}, known as

the fine uniformity for (X,r), has the property that

F(r) is the largest uniformity compatible with r.

Since the infimum uniformity does not behave well in

the analogous way, we do not expect that there is always a

smallest uniformity compatible with a given topology r.

It turns out that there exists such a smallest uniformity

if and only if (X,r) is locally compact. (This is given

as an exercise in Bourbaki [2, exercise 15, p. 25] and

Gal [6 ] calls this the Samuel-Shirota theorem.) The fol­

lowing proposition gives an explicit construction of the

smallest uniformity.

Proposition 1. Let (X,r) be a locally compact regular

space. Let U = {U C X x X| 3 compact C C X with

A U (X \ C) x (X \ C) C Int(U)}. Then u is a unifor-

mity for X, t (u ) = r, and any uniformity compatible with

t contains u. 22

Proof. The first four properties of a uniformity are easily satisfied by u . So we must show that given

e u, there is a U^ e u such that U 2 O U 2 C U^.

Take a compact set with (X \ C^) x (X \ C^) C U^.

Take a closed compact set C 3 with Cl Int(C2 ) and then take a compact set C 3 with C 2 C Int(C3).

(C^rj^ ) is compact and regular and hence is uniformiz- able, admitting a unique uniformity v. C Int(U^) so n (C3 x C 3 ) e v. Take V e v such that

V O V C and VtC^] C C2 , and V = V"1 . Let =

V U (X \ 0 3 ) x (X \ C 2 ). We assert that U 2 O U 2 C U^.

Given (x,y),(y,z) e U 2 there are three cases:

Case 1. x e C^. Then (x,y) e V, so ye 0 3 . That implies that (y,z) e V, and we have (x,z) e V O V C U^.

Case 2, z e C^. Then (y,z) e V, so (z,y) £ V and so y e C3 . Hence (x,y) e V, and we have (x,z) e as above.

Case 3. Otherwise, (x,z) e (X \ C^) x (X \ C^) C U^.

(X \ C 2 ) x (X \ C 2 ) C Int(U2 ) .since C 2 is closed.

This also tells us. that \ c C IntCl^)* t^ie other hand, AP is in the interior of V when this interior 2 is taken relative to C 3 . So there is an open set

0 C X x X such that A^ C 0 n (C^ x C3 ) C V. But

C 2 C lnt(C3) so Ac C 0 n (Int(C3) x Int(C3)) C V, and hence A^ C Int(V). Then A^ C IntCl^) since 23

AX = ^C2 1-1 AX \ C2* ^hus ^2 G u* ^ow we ^ave shown that u is a uniformity.

To show r(j/) C r, suppose x e 0 e r(y). There

is some U e u with U[x] C 0. Take 0 e r with A (x,x) e 0 x 0 C U. Then x e 0 = (0 x 0 )[x] C X X X X X u[x] c 0 .

To show t c t (u ), suppose x e 0 e r. Take an open set 0* e T with cO* compact, and x e 0“ , cO* C 0. Let U = (0 x 0) U (X \ cO*) x (X \ cO*) e U.

Then x e U[x] = 0.

Given any uniformity u* with t(u * ) = t we assert

that U C u*. Given U e u, there is some compact

C X with (X \ C^) x (X \ C^) C U. Take a compact

set C 2 with C Int(C2 ). We have ^ = ^I c 2

since (^2 ,rlc2 ^ admits a unique uniformity. So there

is a U* e u* with U* n (C2 x C2) C U, U*^] C C2,

and U* = IT-1. We assert U* C U. Given (x,y) e U*

there are three cases:

Case 1. x e C^. Then y e C2 so (x,y) e U“ n

(C2 x C2) C U.

Case 2. y e . Then x e C2 since (y,x) e U",

so (x,y) e U as above.

Case 3. Otherwise', (x,y) e (X \ C^) x (X \ C^) C U

Since U* C U , we must have U e u*. So we have

shown that u C u* . Q.E.D. 24

This construction is different from that outlined in

the statement of the exercise in Bourbaki mentioned above, and from those of Samuel [14] and Shirota [16], and it will be useful later. From now on we shall refer to the

smallest uniformity compatible with a given locally compact regular topology as the cofine uniformity. Various characterizations of the cofine uniformity are given by

the following proposition.

Proposition 2. Let (X,t/) be an arbitrary uniform space, and let (R,f/) stand for the reals with the usual metric uniformity. Then the following statements are equivalent.

Furthermore, if (X , £/) satisfies any of these properties then it is locally compact and totally bounded.

1. (X,t/) is minimal, i.e., if u* is a strictly smaller uniformity for X, then r(t/*) ^ r(t/).

2. If S : D ■+ X is a net with no cluster points

then S is Cauchy.

3. If F is a filter on X with no cluster points then F is Cauchy.

4. There is a compact space (Y,v) where (X,y) C

(Y,v) and Y \ X is a singleton.

5. Given a map f: (X,£/) (R,w) and e >0, there is a compact set C C X and a map g: (X,£/) -*• (R,w) such that g is constant on X \ C and |f(x) - g(x) | < e for each x e X. 25

6 . Given a map f: (X, i;) (R, w) and e > 0, there is a compact set C C X such that for any x,y e X \ C we have |f(x) - f(y)| < e- 7. For any two closed sets A,B C X, if there is a U e v with U[A] n B = 0 then either A or B is compact.

8 . If u* is a uniformity for X which is strictly smaller than £/, then either (X,t/*) is compact or f“l u" is strictly larger than n t/* 9. For each U e u there is a compact set C C X with (X \ C) x (X \ C) C U.

10. (X,tf) if cofine, i.e., if U* is a uniformity for X with t ( u*) = t ( u ) then u O v * m

Before proving the proposition some discussion is in order. First of all it should be noted that if (X,w) is compact then all ten properties are trivially satisfied.

Therefore, in the following proof we shall make liberal use of the phrase "without loss of generality, (X,u) is not compact."

Secondly, the class of spaces which admit unique uniformities is larger than the class of compact spaces, and has been investigated by Doss [5], Newns [13], and

Gal [6 ], and their results have been summarized by

Warren [19]. Doss showed that uniformities which are uniquely compatible with a given topology satisfy properties 26

3 and 4 and a property similar to 7, Newns showed that such spaces satisfy property 9 and something similar to

8 , while Gal showed that they satisfy a property similar to 5. By assuming Proposition 2 together with the standard properties of the fine uniformity, we will give new proofs of some of the results of Doss, Newns, and

Gal.

Thirdly, it is evident from the statement of the exercise in Bourbaki that the equivalence of properties

4 and 10 and the fact that such spaces are locally compact and totally bounded was known. Property 4 says that u is the uniformity inherited by X from its

Alexandroff compactification, and this fact was central to the proof in Samuel [14] of the Samuel-Shirota theorem.

Dieudonne [4] used a proof that 9 implies 4 to show that the deleted Tychonoff plank (see Stean [17]) admits a unique uniformity. The rest of Proposition 2 appears to be new, especially the fact that minimal uniformities must be the cofine uniformity.

Fourthly, properties 2 and 3 have a special signifi­ cance in that any net (filter) which is convergent is

Cauchy, and any net (filter) which has cluster points but no limit points is not Cauchy. So the only nets (filters) for which the Cauchy property is not decided by the topology are those which have no cluster points. In a sense then properties 2 and 3 say that any net (filter) which could be Cauchy, is Cauchy.

Two lemmas will be used in the proof of Proposition 2

Lemma 2.1. Let u and u* be uniformities for X.

Assume y* C £/, that V satisfies property 7, and that

(X,y*) is not compact. Then for any x e X there is a

U* c y* such that U*[x] is r( y)-compact. (In particu­ lar, by letting y* = y, we get that property 7 implies that (X, e/) is locally compact.)

Proof. If every r(y*)-neighborhood of x had a compact complement, then (X,y*) would be compact. So we can take

0 e T (t/*) with x e 0 and X \ 0 not compact. Take U* G u* with U* closed and U* O U*[x] C 0. Then

X \ 0 and U*[x] are r(y)-closed, and U*[U*[x]] n

(X \ 0) = 0, and U* e u. X \ 0 is not r(u)-compact so

U*[x] is r(u)-compact.

Lemma 2.2. Property 7 implies that (X,y) is totally bounded.

Proof. Suppose that (X,y) is not totally bounded. Then there is a sequence S: N ■+■ X and a U e u such that for any n ^ n we have (Sn j5>m) i U. Let S^ and S^ be the subsequences of even and odd terms of S. Take a symmetric V e u with V O V O V C U. We have c(S^[N])

C V[S°[N]], and c(S1 [N]) C V[S 1 [N]]. So 28

V[c(S°[N])] n c(S1 [N]) = 0 since U[S°[N]] n S1 [N] = 0. By property 7 either c(S 0 [N]) or c(S 1 [N]) is compact.

0 1 Then either S or S has a Cauchy subnet and thus S has a Cauchy subnet. But this contradicts the choice of

S; if f:D -*■ N is such that S O f is Cauchy, there must be a d e D with S O f[d,°°) C U[S(f(d))]. But then

“ {f(d ) } and so f is not cofinal.

Proof of Proposition 2. Since the lemmas have shown that property 7 implies that (X,t/) is locally compact and totally bounded it remains to show that the ten properties are equivalent.

1 implies 2. Given a net S: D X with no cluster points and assuming that u is minimal, it suffices to find a uniformity t/* such that u" C u , r(t/*) = r(t/), and S is £/*-Cauchy.

For p e D and Vet/ let R(p,V) = U {V[Sm]xV[Sn] | m,n £ p}. Let u* = {U e t/| R(p,V) C U for some p e D, Vet/}. The first four properties of a uniformity are satisfied by u * t so we must show that given e v* there is a e u* with U 2 O l>2 C . There is some R(p,V) C U1. Take Wet/ with W O W C n V and

W = W"1 . Let = W U R(p,V). We assert that U 2 satisfies the above property. Given (x,y),(y,z) e U 2 there are four cases: 29

Case 1. (x,y),(y,z) e W . Then (x,z) e W OW C U^.

Case 2. (x,y),(y,z) e R(p,W). Then x e W[Sm] and

z e ^tSn] for some m,n 2 p, so (x,z) e R(p,W) C

R(p,V) C Ux . Case 3. (x,y) e R(p,W), (y,z) e W. Then x e W[Sm ], y e W[Sn], and z e W[y]. So (x,z) e x W O W [Sn ] C

R (p ,W O W) C R(p,V) C 1^.

Case 4. (x,y) e W, (y^z) e R(p,W). Then x e W^ C y ]

= W[y] and y e W [ S j , z e W[Sn ], So (x,z) e W O W [ S j x

W[S ] C U. as above, n 1

Since U2 e u* we have shown that u* is a uniformity.

To show that r(£/) C r(u*) suppose x e 0 e r(u).

Since x is not a cluster point of S , we can take U e u

— 1 and p e D such that U[x] C O , U = U , and for any n 2 p we have S^ i U[x], Then for each n 2 p, x I U[S ], so R(p,U)[x] = 0. Since U U R(p,U) e u* and (U U R(p,U))[x] = U[x] C O , we have shown that

0 e r(u*).

It is clear that un C u, so we have r(t/*) = r(i/).

It is also clear that S is u*-Cauchy, since given U e u*

there is an R(p,V) C U, and so whenever m,n 2 p then

(S.Sjm 7 n' e U. 2 implies 3 . This is a standard net-filter argument.

Given a filter f on X with no cluster points, let 30

D = {(F,x)| x e F e f} . Define (F^,x^) 2 (F^,X2 ) if and only if F^ C F 2 * Let S: D ■+ X be defined by

S(F,x) = x. Then if x were a cluster point of S, x would be a cluster point of F, since S[d,<=°) = F when d = (F,x) e D and so if x e cS[d,°°) for each d e D then x e cluster(F). So S has no cluster points.

Then by property 2, S is Cauchy. This implies that given U c U there would be some F e f such that for any x^, X2 e F, (x^,X2 ) e U. But in that case F x F C U.

So f is Cauchy.

3 implies 4 . Without loss of generality, (X,u) is not compact. By Lemma 0.4 (X, u) has a completion (Y,v), which we will regard as containing (X, v) as a subspace, with a certain separation property: given y^ and ^2 to be distinct points in Y which are not both in X, there is a V e V such that V[y^] n V[y 2 ] = 0* Assuming that (X, u) satisfies 3, we will show that

(Y, v) is compact and Y \ X is a singleton. To show

(Y, v) compact, it suffices to show (X, u) is totally bounded since (X, u) is dense in (Y, v) , and so (Y, v) would be totally bounded and complete, hence compact.

Given an ultrafilter F on X, either f converges or has no cluster points. In either case F is Cauchy, and this shows that (X, u) is totally bounded. 31

Since (Y,v) is compact we know that Y \ X f- 0, since we are assuming (X,t/) not compact. To show Y \ X is a singleton, suppose e Y \ X. Take f^ and to be filters on X converging to and respectively. F^ and F^ do not cluster to any point in

X because of the separation property of (Y,v). Then

F 1 ^ F 2 a filter which does not cluster to any point of X, and so F^ n F^ is Cauchy. Supposing that

^1 ^ ^ 2 f t^iere woufd be a closed V e V such that

^ Take F e Fi 1-1 F 2 wfth F x F CIV. Then since V is closed, cF x cF CZ V. But e so

^1^2^ e Y, a contradiction. So y^ = y^. 4 implies 5. Assume that (X,tO is not compact,

(X,tf) C (Y,v), (Y,v) is compact, Y \ X = {y*}> f: (X, £/) (R,w), and c > 0. Since (X,*/) is not compact it must be dense in (Y,v), so (Y,v) is a completion of

(X,u). Hence the map f has an extension f: (Y,v)

( R , w ) . Take 0 e r ( i / ) with y* £ 0 and for each x e cO,

|'f(x)-lf(y*)|

(Y,v) is compact and regular, hence normal, so we can apply the Tietze extension theorem. Take h: (Y ,r(v)) -*■

( R ,r ( & r) ) such that h | cQ = 7(y*) - T \cq and j.h(x)|< e for each x e X. Since (Y,v) is compact and h is continuous, we have h: (Y,v) (R ,(v ). Let g: (X,i/) -*■ ( R ,w ) be defined by g(x) = f(x) + h(x). Then for x £ C, g(x) = 7(y*) and for all x e X, |g(x) - f(x)| = |h(x)| < e. 32

5 implies 6 . Given f: (X,t/) (R,fi/) and e > 0, take C compact and g: (X,£/) ■+■ (R,fv0 so that

|g(x) - f(x)[ < % e for each x e X and g(x) = r for x i C. Then if x,y £ C, we have |f(x) - f(y)] £

I f (x) - r| + |r - f(y )| = |f(x) - g(x)j + |g(y) - f(y)| < e.

6 implies 7. Given closed sets A,B with

U[A] n B = 0 for some U e u, by Lemma 0.3 there is a map f: (X , u) (R,w) such that f[A] C {0} and f[B] Cl

{1}. By property 6 there is some compact set C such that for x, y £ C we have |f(x) - f(y)| < 1. We assert that either A C C or B C C; otherwise take x e A \ C and y e B \ C and then f(x) = 0, f(y) = 1, but |f(x) - f(y)I < 1. So either A is compact or B is compact.

7 implies 9 . Assume that (X,u) satisfies 7 and is not compact. Then by Lemma 2.2, (X,u) is totally bounded and noncompact, hence non-complete, and hence not uniformly locally compact. So for any We u, there is some x e X where W[x] is not compact. We use this together with 7 to prove 9. Given U e tf, take V e u with V = V"1 , V O V C U, and V open. Take Wet/ with W 0 W C V and W closed. Take x e X with W[x] not compact. Then W[x] and X \ V[x] are closed sets, and W[W[x]] C V[x]. By 7, since W[x] is not compact,

X \ V[x] must be compact. Since V[x] x V[x] C V O V C U we have shown 9. 33

9 implies 10. It suffices to show that property 9

implies that (X,tf) is locally compact, since by Proposi­

tion 1 a uniformity satisfying 9 is contained in the cofine

uniformity.

Assume that (X,tf) satisfies 9 and is not compact.

Let f be the filter on X defined by f = {F C X|

3 compact C C X with FUC=X}. By 9, f is Cauchy.

If x is not a point of local compactness of X, then x

is a cluster point of F, since no neighborhood of x is

contained in a compact set. So we have that F converges

to x, since a Cauchy filter converges to its cluster points. But this means that every open set containing x has a compact complement, which implies that X is compact, a contradiction. So X is locally compact.

10 implies 1. Trivial. So far we have shfown that properties 1-7, 9, and 10 are all equivalent. So it remains to relate these to property 8 .

1 and 7 imply 8 . Assume u* C u, (X,u*) is not compact, (X,£/) satisfies 7, and n u“ - Hi/, and we will show that r(cr*) = t(u ) . Then assuming 1 gives

v* = u which proves 8 .

Here, where we have two uniformities, the symbol c* will mean closure with respect to u * t and c will be

closure with respect to u. 34

Given A C X, a r(t/)-closed set, to show that A is r(t/*)-closed. Given x e c*A, take U* e u* such that

U*[x] is r([/)-compact (using Lemma 2.1). A n U*[x] is r(EE)-compact since it is the intersection of a t (u )- closed set with a r(i/)-compact set, hence A fl U*[x] is

T(u* ) -compact. Then we have c*(A fl U"[x]) =

(n t;*)[a n u*[x]] = (n e/)[a n u*[x]] = c (a n u*[x]) c a , by Lemma 0.1. But x e c*(A n U*[x]) since given V* e u~{ we have V*[x] n U*[x] n A = (V* n U*)[x] n A ^ 0.

So x e A, and we have shown that A is r( £/*)-closed.

8 implies 1. Assume (X,u) is not compact. By property 8, if u* is strictly smaller than i/, then either

(X,i/*) is compact or n u* £ PI u. If (X,t/*) is compact

then t (u*) f r(t/). On the other hand, if (x,y) e

(De/*) \ (nu) then y e c"{x} \ c{x},' so r(t/*) ^ t (u ).

Q.E.D.

Using Proposition 2 we can give short proofs of some of the characterizations of spaces with unique uniformities.

We will need to use the following property of the fine uniformity: if (X,w) is fine, and f: (X, r(u)) -*■ (Y,r(t/)) then f: (X,v) -*■ (Y,v).

Proposition 3. [Doss] A necessary and sufficient condi­

tion that a uniformizable space (X,r) admits a unique uniformity u, is that whenever A and B are closed normally separable sets then either A or B is compact. 35

Proof of Necessity. Let d be the usual metric on the

reals, and let f: X -*■ R be a continuous map with

f[A] C {0} and f[B] C {l}. Since u is unique it is both fine and cofine. Since u is fine, f is uniformly

continuous with respect to u and u Let U =

(f x f)- 1 [V, i] e u. Then U[A] fl B = 0. Since V is a , ^ cofine then by Proposition 2 either A or B is compact.

Proof of Sufficiency. If we show that an arbitrary u compatible with is cofine, then this will mean that u is unique. By Proposition 2, it is sufficient to show

that (X,£/) satisfies property 7. Given closed sets

A,B C X and U e u such that U[A] n B = 0, by Lemma

0.3 we have that A and B are normally separable. So by Doss's condition either A or B is compact, and this proves property 7. Q.E.D.

Let C(X) be the set of continuous real-valued

functions on X, endowed with the topology of uniform convergence, and let A(X) be the subset of C(X) con­

sisting of those functions which are constant on the complement of some compact set.

Lemma 4.1. For any space (X,u), the members of A(X) are uniformly continuous. Proof. Given g e A(X) there is some compact C C X

such that gix ^ is constant. Given e > 0 it suffices 36

to find U e u such that if (x,y) e U then

|g(x) - g(y)| < e. Since g is continuous, for each x g X we may choose a U A e u such that whenever y e U O U [x] then |g(y) - g(x)| < %e. The collection X A {Xnt(U [x])| x g C} covers C, so there is a finite set F C C where {U [x]| xgF} covers C. Take a symmetric

U g U with U C n(U IxeF). Then we assert that if

( x,y) G U then |g(x) - g(y)| < e. For if x,y g X \ C then g(x) = g(y). On the other hand, suppose x g C. Then x g U .[x']» Cot' some x' e F. Then y e U[x] C A U O Ux ,[x'] C Ux , o Ux ,[x']. So |g(x) - g(y)| £

|g(x) - g(x’)| + |g(x') - g(y)| < hz + %e = e. If instead we assume y e C, we get the same result since U is symmetric.

Proposition 4. [Gal] A necessary and sufficient condition that a uniformizable space (X,r) admits a unique uniformity u is that A(X) is dense in C(X). Proof of necessity. u is both fine and cofine. Since it is fine, all of the members of C(X) are uniformly con­

tinuous. Since u is cofine, (X,u) satisfies property 5 of Proposition 2, so A(X) is dense in the set of uniformly continuous functions on (X,u). Hence A(X) is dense in C(X).

Proof of sufficiency. Suppose A(X) is dense in C(X).

Let u be any uniformity compatible with T. By Lemma 37 4.1, the members of A(X) are uniformly continuous with respect to u. So any map f: (X,t/) (R,w), being a member of C(X), can be approximated arbitrarily closely by map g: (X,i/) -*• (R,w) in A(X): this says that (X,t/)

satisfies property 5 of Proposition 2. So any u compatible with t is cofine. Hence there can be only one such u. Q.E.D.

Proposition 5. [Newns] A necessary and sufficient condi­ tion that a uniformizable space (X,r) admits a unique uniformity is that for any u compatible with r, for any

U e u, there is a compact set C C X with (X \ C) x (X \ c) c u. Proof. In view of property 9 of Proposition 2, Newns' condition says that a space admits a unique uniformity if and only if each uniformity for the space is cofine.

Q.E.D. CHAPTER 2

PROPERTIES OF THE COFINE UNIFORMITY

In this chapter we look for invariance properties of the cofine uniformity. We find both positive and negative results. All of the proofs here are applications of

Proposition 2.

Proposition 6 . Given a suijective map f:(X,y) (Y,v), if u is cofine then v is cofine.

Proof. Given a net S:D -*■ Y with no cluster points, we can choose a net S*:D ■+ X such that f O S” = S, since f is onto. S* has no cluster points since f is con­ tinuous. So S* is Cauchy and that implies that S is

Cauchy, since f is uniformly continuous. Q.E.D.

Proposition 7. Given a map f:(X,y) (Y,v) where f[X] is closed, v is cofine, and u is weak for f and v, then u is cofine.

Proof. Given a net S:D ■+ X, if f O S has a cluster point then S does. This is because if y is a cluster point of f O S then y e f[X], so we may choose x e X with f(x) = y; then x will be a cluster point of S, since given d-^ c D and U e u we can take V e v such

38 39

_ 'J that (f x f) [V] C U and we can take d 2 £ D such that d 2 s dj and d 2 e (f O S)_1 [V[y]] = S_1 [f"1 [V[y]]]=

S_1[((£ x f)”1 [V])[x]] C S_1 [U[x]]. So if S has no cluster points then f O S has no cluster points, and is therefore Cauchy. So S is

Cauchy, since given U e u we may take V e v with

(f x f)‘1 [V] C U and then take d £ D such that if d^,d 2 £ d then (f O S(d^),f O S(d2 )) £ V. But -this last inclusion is the same as saying that (S(d^),S(d2 )) £

(f x f)"1 [V] C U. Q.E.D.

Corollary 7.1. If (X,£/) is a closed subspace of (Y,v) and V is cofine then u is cofine.

This last invariance property of the cofine uniformity stands in contrast to fine uniformity, which when res­ tricted to closed subsets of a normal space is fine, but not necessarily when restricted to closed subsets of a non-normal space.

Example 1. The deleted Tychonoff plank, [0,u^] x [P >“o ] \

(w ^,cJq ), admits a unique uniformity (see Dieudonne [4] and Newns [13]) and so this uniformity must be both fine and cofine. The subspace x [0»w q ) inherits a uniformity which is cofine but not fine. This subspace is closed, and is homeomorphic to N, the set of natural numbers with the discrete topology. The cofine uniformity 40 on N is the so-called flag uniformity, which has for a basis the collection of sets of the form U = {(n,m) | n = m or n,m £ p}. The fine uniformity on N is the discrete uniformity.

Proposition 8 . If (X,t/) = x {(X^,u ^ | a e L} then u is cofine if and only if either 1 ) each is compact, or 2 ) some u ... is cofine and for each a r a*, u is a" indiscrete.

Proof. Suppose (X,t/) is cofine. Then each •*-s a projection of (X,t/) and hence cofine by Proposition 6 .

Suppose (XaVf» ^a*) is non-compact and ua £ {XQ x X a } for some a ^ a*. Take U e u with U ^ X x X and a a - a a a then take (x1 ,x0) e X x X \ U . Let U = (P x P )~^[U I, ' 1 * 2 ' a a ' a a a a 7 A = P -'l [x-1, B = P — 1 [x0], where P is the a-projection oi 1 a L 2 J 7 a r j map. cA and cB are non-compact since Pa*[cA] =

P a" .--[cB] *■ J = X a" Take V e u with V =* V**^ and V O V 0 V C U. Then since U[A] n B = 0, we have

V[cA] (“1 cB = 0, a contradiction.

Conversely, if case 1) is true then (X , t/) is compact and so cofine, and if case 2 ) is true then u is weak for (X,t/) (X * , 0 Q*) and hence cofine by

Proposition 7. Q.E.D.

Proposition 9. Given two spaces (X^,£/^) and (X2 ,^2 ^ with (X,t/) = (X^,u^) + (X2 ,U2 ) then u is cofine if and 41 only if one of the summands is compact and the other cofine.

Proof. Suppose u is cofine. X^ and X 2 are closed subsets of X, so and t/2 are cofine by Corollary

7.1. Let U = x ^ U X 2 x X 2 e E/. Then U[X1] n

X 2 = 0 , so either X^ or X 2 is compact.

Conversely, suppose i-s compact and (X2 ,e/2) is cofine. Given a net S:D -*■ X with no cluster points,

S must eventually be in X2, hence S is Cauchy. Q.E.D.

In contrast to the last result, the union of two cofine uniform spaces does have the cofine uniformity when the intersection of the spaces is "big enough."

Proposition 10. Let X = Xt U X 2 with (X^t^) c (X,u) and (X2 ,u2) C (X,tf), and assume X^ n X 2 does not have compact closure. Then if u^ and u^ are cofine so is u.

Proof. We use property 6 of Proposition 2. Given a map f:(X,u) (R,w) take closed compact sets C^ C X^ and

C 2 C X 2 such that if x and y are both in X^ \ C^ or both in X 2 \ C 2 then |f(x) - f(y) | < %e. (We can choose C^ and C 2 to be closed since the closure of a compact set is compact in a uniform space.) Now X^ l~l X 2 cannot be a subset of C^ U C2. So take z c (X^ n X2)\

(C^ U C2). Then whenever x,y e X \ C^ L) C 2 we have 42 j f(x) - f(y)| £ |f(x) - f(z)| + 1 f(z) - f(y)| < + hz =

0 . Q.E.D.

In the rest of the chapter we will try to justify the term "cofine" from the point of view of category theory.

As pointed out before, the fine uniformity has the property that if (X,t/) is fine then all continuous maps from (X,u) are uniformly continuous. We might expect that the cofine uniformity satisfies the dual of this property, but it turns out that a restriction must be placed on the kind of maps involved.

Proposition 11. If f:(X,r(i/)) -*• (Y,r(v)) is a proper map and u is cofine, then f:(X,tf) (Y,v), i.e., a con­ tinuous proper map to a cofine space is uniformly continuous. Proof. Since (Y,r(v)) is locally compact then so is

(X,r(y)). Given V e v take C to be a closed compact set with (Y \ C) x (Y \ C) C V. (We may assume C is closed since the closure of a set C as in property 9 of

Proposition 2 will still satisfy that property.) Let

C' = f"1 [C] a compact set in X. Then (X \ C') x

(X \ C') C (f x f)"1 [V]. Take a compact set C ,f C X such that C* C IntC''. Then since f|^« « is uniformly continuous we take U e u such that U fl C 1' x C' ' C

(f x f)“1 [V], U[C'] C C " , and U = iT1 . Then .43

U C (f xf)"^[V]. For if (x,y) e U there are three cases:

Case 1. x c G1. Then y e U[x] C C ' 1 so (x,y) e

U n C'' x C'' C (f x f)“1[V]. Case 2. y e C1. Then x e U[y] C C' ' so (x,y) e u n c'’ x c'' c (f x f)”1[v]. Case 3. Otherwise (x,y) e (X \ C') x (X \ C 1) C

(f x f)"1 [V], Q.E.D.

The following example shows that we cannot drop the properness of the map from Proposition 11.

Example 2. Let Y be the unit circle in the complex plane and let X be Y \ {-l}. Give both the usual metric uniformity. Let f:X ■* Y be given by f(x) = x ^ ^ , taking the principal value of the square root. Y is endowed with the cofine uniformity since it is compact.

(Note that X also has the cofine uniformity.) f is continuous but not uniformly continuous, since points arbitrarily close to - 1 are separated by f according to whether their imaginary parts are positive or negative.

Note that f is not proper since f ^[Y] = X.

The next result shows that under certain conditions the properness of the map is necessary for uniform continuity. Proposition 12. Given £:(X,y) ■* (Y,v) with u cofine

and t ( v) locally compact, then f is proper if f[X]

does not have compact closure in Y.

Proof. Suppose f is not proper. Take a closed compact

set C O Y such that A = f —11 [C] is not compact. Take

also an open relation V e v such that c(V[C]) is com­

pact and let B = f“^[Y \ V[C]]. Then A and B are closed in X. Let U = (f x f)"^[V]; then U[A] n B = 0 so B is compact. Then f[B] = f[X] \ V[C] is compact

and so is its closure. We reach the contradiction that

f[X] has compact closure, since f[X] C c(V[C]) U

c(f[B]). Q.E.D.

Proposition 11 has an interpretation in category

theory which can be contrasted with above mentioned property of the fine uniformity. We must define the * following categories and functors:

Let CR be the category of completely regular spaces

and continuous maps; let U be the category of uniform

spaces and uniformly continuous maps; let LCR be the category of locally compact regular spaces and continuous proper maps; and let LCU be the category of locally compact uniform spaces and uniformly continuous proper maps.

Let f:CR -*■ U be the functor which takes (X,r) to

( X , f ( t ) ) where f ( t) is the fine uniformity for T; let w:LCR LCU be such that w(r) is the cofine uniformity; 45- and let r:LCU -*■ LCR or U + RC be such that r(t/) is the topology generated by u. Each of these functors leaves the maps alone, i.e., the image of a morphism is the morphism which is actually the same map as the original morphism.

We note that the properties of fine and cofine uniformities guarantee that F and M really are well- defined functors, that given a morphism f in CR or

LCR, the morphism F(f) or w(f), which is actually the same map as f, is uniformly continuous and hence in U or LCU, respectively.

Proposition 13. F is left-adjoint to z’iU -*■ RC and M is right-adjoint to Z’tLCU LCR. The adjunctions .

n^LSu( fX,Y) + CR(X, ZY) and n 2 :LCR(z’X,Y) LCU(X,WY) are such that U-^Cf) and 112(f) are same map as f. Proof. From Proposition 11 and the discussion preceding it, we know that and 112 are well-defined bijections. It remains to show they are natural transformations, but this turns out to be a trivial consequence of the way they are defined.

Given a proper map f:(X2 ,tf2) ^ l > Ul) an<* a ProPer map g:(Y^^) *► (Yj,r2) we assert that the diagram in

Figure 1 commutes. But this just says that given h: (X.pZ’Ct^)) + (Y1 ,r1) then 46 n 2 (S Oho t f) = Mg o ^ 2 ^ ) O f. This equation is true since, as far as sets are concerned, both sides are the map g O h O f. The naturality of r\ ^ works the same way. Q.E.D.

n2 LCR(rX1 , Y-^) LCU(X1 »mY1)

(rf,g) (f,Mg)

LCR(rX2 ,Y2) ■> lcu(x2 ,my2)

FIGURE 1.

A Commutative Diagram for Adjointness of m CHAPTER 3

THE HYPERSPACE UNIFORMITY

It is traditional to trace the interest in hyperspaces to Hausdorff [7], who put a metric on the set

of closed bounded subsets of a given metric space.

Bourbaki [1, Chapter 2, Sec. 2, Exercise 7, p. 145]

introduced a uniformity on the closed subsets of a uniform

space, and this construction agrees with the Hausdorff metric in the case when the given uniformity is metriz- able.

A thorough introduction to the basic properties of

the hyperuniformity was done by Michael [12]. Further work on the hyperspace uniformity was done by Caulfield

[3], and alternate definitions of uniformities on the hyperspace were given by Scott [15], Hovis [8 ], and

Levine [11]. The construction of the standard hyper-uniform space goes as follows. We are given an arbitrary space (X,t/). v Let 2 be the set of non-empty closed subsets of X.

Given a relation U C X x X, let H(U) = {(A,B) e 2^|

A C U[B] and B C U[A]}. Then {H(U)|U e u } is a basis

47 48 v for a uniformity on 2 . We denote this uniformity by

2 U. Then (2^,2U) is the hyperspace of (X,tf), and

(X, £/) is called the base space of (2 ^, 2 ^). X ii We note at this point that (2 ,2 ) is always a

Hausdorff (T^) space. We will be making use of the following definitions which we borrow from Caulfield [3].

Definition. Given a net S:D 2^, let

lim S = {x c X| V O e t (u ) , if x e 0 then V d e D,

3 d' £ d with n 0 0 0 }

lim S = {x e>X| V O e t (u ) , if x e 0 then 3 d e D with Vd' U , Sd» n 000}

Proposition 14. a) lim S and lim S are closed subsets of X.

b) lim S (Z lim S y c) If A e 2 is a cluster point of S, then

lim S C A C lim S

d) For d e D let = ujs^ild' £ d}. Let Fg

be the filter on X generated by the filterbase

{F^|d e D}. Then lim S is the set of cluster points of

F g , and lim S is the set of points x with the property

that for any . S*, a subnet of S, x is a cluster point

O f F g * . 49

Proof. Proofs of a) and b) are straightforward. We will start with c) and show lim S C A. Since A = cA =

n {U [ A ] [ U e £/} = D {U" 1 O U[A]|U e tf}, it suffices to show that lim S C U“^ O U[A] for arbitrary U e u. .

Given x e lim S there is a d e D such that for any d' 2 d we have S^i f~l U[x] f 0. Choose a d* 2 d so that e H(U)[A]. Since S^* f“l U[x] £ 0 we have x e U"1 [Sd*] C U " 1 0 U [A] .

Next we show A C lim S. Given x e A, and

0 e t ( u) with x e 0, and d e D, take U e u with

U-'*'[x] C 0, and take d 1 2 d such that S^i e H(U)[A].

Then x e A C U[Sd t], so S^i PI U_^[x] ^ 0, and there­ fore S^t n 0 £ 0 . Now we show that lim S is the set of cluster points of Fg. The statement x e lim S is equivalent to saying that for any 0 e t( u) with x e 0 and for any d e D, we have n 0^0, and this is equivalent to x e n {cFd |d e D }, i.e., x is a cluster point of f g-

To prove the last part of d) , we first show that if x e lim S and S* is a subnet of S, then x is a cluster point of ^g*- We assume that S* = S O f, where f:D* ■* D is a cofinal map of directed sets. Given

0 e ?(u) with x e 0 there is a d e D such that for any d* 2 d, S^i n Oj^0. Take d* e D" so that for 50 any d*' 2 d* we have f(d*‘) 2 d. Then for d*' 2 d*,

S*^*i n 0 ^ 0 ; so x isa cluster point of Fg*.

Finally, we assume x i lim S and we will find a subnet S* of S where x is not a cluster point of

FgVr. There is an 0 e t ( u) with x e 0 and for any d e D, there is a d' 2 d with S^t n 0=0. Let D* = {d c D | f“) 0 = 0} and let f:D* -*■ D be the inclusion map. Then f is a cofinal map of directed sets. Let S* = S o f ; then x is not a cluster point of Q.E.D.

We will also find useful the following lemmas which can be found in Michael's paper [12].

Lemma 15.1. If A e 2X then (2A, 2UIA) C (2X ,2t/).

y Lemma 15.2. If A C 2 and A is compact then V U A e 2 (assuming A ^ 0), i.e., a compact union of closed sets is closed.

y Lemma 15.3. If A C 2 and A is compact and each

A c A is compact, then U A is compact, i.e., a compact union of compact sets is compact.

Lemma 15.4. If A is closed in X then {B e 2X |B C A} V is closed in 2 . 51 y Lemma 15.5. If A is compact in X then {B e 2 |

B (1 A / 0} is closed in 2^.

Lemma 15.6. Let f:X ->2 be defined by f(x) = c{x}.

Then f[X] is closed in 2^ and is the TQ-identification of X.

Proof of 15.1. Since A is closed in X then 2^ C 2^.

Moreover, H(U n A x A) = H(U) f“l 2^ x 2^ for each

U e u. So 2^1 A = 2 U\ 2A.

Proof of 15.2. Let S:D + UA be a net which converges to x £ X. Let S*:D -*■ A be chosen so that S^ £ S*d for each d £ D. There is an A £ A which is a cluster point of S*. We will show x £ A C U A by showing x £ U[A] for arbitrary U e Uf and noting that A = ca = n {U [ A ] | U e £/}. Take Vet; with o V C U , and take d e D so that for d* 2 d we have S^i £ V[x]. There is a d* 2 d where e H(V)[A]; then we have S^* £

S*d* C V[A]. So x e CI V " 1 O V[A] Cl U[A].

Proof of 15.3. Let S:D ^ U A be a net and choose S* as in the last proof, with A e A a cluster point of S*.

We will show that S has a cluster point in A by showing for each d g D that A n cSLd,°°) j4 0, and thus by the compactness of A that A n cluster(S) f4 0.

Suppose for some d e D that A n cS[d,“] = 0. Then 52

since A is compact, there is some U e u with U[A] n S[d,°°) = 0. Choose d* £ d such that £ H(U)[A]. Then S^* £ UCA], a contradiction since

S c S[d,t») .

Proof of 15.4. Suppose B* G? A. Choose x e B*\ A, and

U g u such that U"^[x] n A = 0. We assert that

H(U)[B*] n {B e 2X|B C A} = 0. For if C e H(U)[B*] then x e B* C U[C], so there is some x' e C with x £ U[x']. So x' e U^ f x ] and therefore x 1 i A, showing that C (£ A.

Proof of 15.5. Suppose B* n A = 0. Then by compactness of A there is a U e U with U[B*] fl A = 0. So

H (U) [ B* ] n {B e 2X |B PI A ^ 0 } = 0 for if C e H(U)[B*]

then C C U[B*].

Proof of 15.6. The T^-identification of X is exactly the set f[X] with the quotient uniformity. So we must show that the quotient uniformity on f[X] agrees with

2 fflf[X]- ®ufc will follow if we show that when U e u is open and symmetric, (x^,X2 ) e U if and only if

(c{x^},c{x2 }) e H(U), for in that case (f x f) ^[H(U)] =

U. If X2 e U[x^] then since U[x^] is open and X is regular we have °{x2 l ^ U[x^]; vice versa, if x^ e

U[x2 ] then c.{x^} C U[x2 ]» Conversely, assume 53

(c {x^ },c {x2 }) £ H(U). Then X2 G U[c{x^}] and so

U[x2] n c {x1 } f- 0 hence x^ e U[X2 3 j likewise X2 G U[x1].

It remains to show that f[X] is closed. Given y A e 2 \ f[X], there are points xi > x 2 £ ^ such that x^ £ c{x2 l* So there is a U e u with X2 fi U[x^].

Take an open symmetric Ve u with V o V C U. We assert H(V)[A] n f[X] = 0. Suppose c{x} e H(V)[A].

Then x^j x 2 e V[c{x}] so c{x} (“1 V[x^] ^ 0 and c{x} n V [ x 2 ] T4 0, and so x e V[x^] n V[x 2 ]. Then

X 2 e V[x]" C V O V[x^] C U[x^], a contradiction.

We now prove one of the most important basic properties of the hyperspace construction. Our proof will follow the outline suggested by Michael [12].

Proposition 15. (2^,2U) is compact if and only if (X, u) is compact.

Proof. Suppose (X, u) is compact. In that case we will first show that the following collection is a subbasis for t(2U) : all sets of the form {A e 2^|A C 0} for y 0 e r(u), together with all sets of the form (A e 2 |

A n 0 ^ 0} for 0 c T(u). These sets are open by

Lemmas 15.4 and 15.5. So suppose we have A* e 0 e r(2y).

Take U z u to be an open relation with H(U)[A*] .C 0

- 1 and take V c u to be open with V 0 V C U. The sets 54 '

V[x] for x e A* cover A*, so take a finite subcover

{V[x^]|k =1,. • •, n). We assert that A* e {A|A C

u [a *]} n {a | a n v[Xl] ?^0}n. • . n {a |a n v[xn ] ^

0} CO. First of all A* C U[A*] and xk £ A n V[xk ]

for each k. Secondly, suppose that A C U[A*] and

A n V[xk] ^ 0 for each k. Then we will show that

A* C U[A] and hence A e H(U)[A*] C 0. Given x e A*, we have x e V[xk] for some k, and since V[xk] n A ^ 0 we have xk e V" 1 [A]. So x e V[xk ] C V o V _1 [A] C U[A].

We have shown that the sets described above form a sub­ basis for r(2u) . X u Next we use Alexander’s lemma to show that (2,2)

is compact by showing that a covering by subbasic open

sets has a finite subcover. Given families of open sets

{0a |a e L^} and {0^ J 3 e L^} such that {{A|A C 0^}| a e L^} U {{A|A n 0^ f 0 } |B e L 2 } covers 2X , let B =

X \ U {0g|3 e L 2 } - Either B = 0 or B e 2^, and in

either case there is some a* e with B C 0a*. Let

C = X \ 0a*; then {0 g|SeL2 } covers C, so let V {On , . . • , On } be a f inite subcover. For any A e 2 1 n either A C 0 or A n 0Q ^ 0 for some k. So k {{A|A C ' {AjA n 0p ^ 0}, . . . , {A|A H 0^ ^ 0}}

y l T1 covers 2 • X u Conversely, suppose (2 ,2 ) is compact. Consider

the map f:X -*■ 2^ defined in Lemma 15.6. Since f[X] is 55

closed in 2^ it is compact. Then since f[X] is the

TQ-identification of X, X must be compact. Q.E.D. CHAPTER 4

LOCAL COMPACTNESS OF THE HYPERSPACE

In this chapter we will try to integrate the material so far by investigating the properties of local compactness and cofineness in the context of hyperspaces.

For a given base space (X,t/) let c(X) be the Y subset of 2 consisting of the non-empty closed compact sets of X. The subspace (c(X),2y| ( 2 ^ , 2 y) has several interesting properties.

Proposition 16. Let (X, u) be locally compact. Then X U c(X) is an open locally compact subspace of (2 ,2 ).

v (This result was misstated in Michael [12] as 2 is locally compact if and only if X is, but it is clear from his proof that the proposition as stated here is . what was intended.)

Proof. Given A e c(X) take B e c(X) with A CZ Int(B) and take U £ u with U[A] C B. Then (2B ,2ylB) CZ

(2X ,2y) by Lemma 15.1, and (2B ,2y lB) is compact by

Proposition 15. Since A e H(U)[A] CZ2B CZ c(X) we may

56 57 say that A has a compact neighborhood which in turn is a subset of c(X). Q.E.D.

Proposition 17. Let (X,t/) be cofine. Then c(X) is X u exactly the points of local compactness of ( 2 , 2 ).

Proof. Given B e 2^ \ c(X) and U e V- we will show Y that H(U)[B] does not have compact closure in 2 .

This suffices since (X,u) is locally compact by y Proposition 2, and thus 2 is locally compact at any point of c(X) by Proposition 16.

Take A e c(X) with (X \ A) x (X \ A) CZ U. Take

0 e t ( u ) with A CZ 0 and cO e c(X). Then B \ 0 is not compact, since if it were then B

(B \ A) C (A n B) U ((X \ A) x (X \ A)[S[d,«)]) C

U[S*d], and S*d C (A n B) U (X \ A) C (A n B) U

C(X \ A) x (X \ A)[B]> since B \ A ^ 0, and so S*d C

U[B].) We will be done if we show that S* hds no cluster y Y point in 2 . Suppose S* clusters to C c 2 . Then ____ J. C CZ lim S“ by Proposition 14c. We assert that lim S" CZ

A. If, on the other hand, x e lim S* \ A, then since x is not a cluster point of S there is some d e D with x f. cS[d,°°). Now x e (X \ S"d ) e but there is no 58 d 1 £ d with (X \ S*^) n S*^* ^ 0» contradicting x e lim S*. So CCA. But if we take Vet/ with

V[A] C O we see that H(V)[C] contains none of the points S*^, contradicting the choice of C. Q.E.D.

Michael [12] and Caulfield [3] defined a way of lifting a map of base spaces (F:X -*■ Y) to a map of the respective hyperspaces (F:2 X 2 Y ) , and showed that this construction preserves uniform continuity. Since we found proper maps to be significant in Chapter 2, the following result is an interesting generalization of that construction.

Proposition 18. Given f:(X,u) (Y,v) let F:2X -*• 2Y be F(A) = cf[A]. Then F:(2X ,2y) + (2Y ,V). If, in addition, f is proper then F lc(x):C^X ^ * C^Y ^ proper.

Proof. Given V* e V take V e v with V O V C V* and then let U = (f x f)~^[V] e y. We assert that H(U) C

(F x F)-^[H(V*)]. Given (A,B) e H(U) we have cf[A] C

V[f[A]] and cf[B] C V[f[B]], so it suffices to show f[A] C V[f[B]] and f[B] C V[f[A]]. Given x e A, there is some x' e B with (x*,x) e U. Then

(f(x'),f(x)) e V so f(x) e V[f[B]]. The proof that f[B] C V[f[A]] works the same way. It is clear that F[c(X)] C c(Y) since the f-image of a compact set is compact and the closure of a compact 59 set in a uniform space is compact. Suppose f is proper. Given a closed compact C C c(x) with C ^ 0 let B = UC, Be C(X) by Lemmas 15.2 and 15.3. Let

A = f”^[B]; then if A = 0, F”^[C] is empty and thus compact, and if A ^ 0 then A e c(X) since f is continuous and proper. 2^ is a compact subspace of c(X) by Lemma 15.1 and Proposition 15. We assert F ^[2®] C

2^; for if A 1 e 2^ is given such that cf[A'] e 2®, then f [A ’ ] C B« so A' C f"^[B] = A. Since C

If (X,t/) is locally compact then in view of

Propositions 1 and 16, c(X) admits a cofine uniformity m( t(2u\ )) * We construct a basis for this unifor­ mity, which depends only on r(£/).

For A d X and R C X x X, let l^(A,R) = {(B,C) e c(X) x c(X)| if either B C A or C C A then (B,C) e

H(R) }. Let Bx = {l/(A,R)|A e c(X) and Ax C Int(R)}.

Then we have:

Proposition 19. If (X,t/) is locally compact then Bx is a basis for w(r(2 U| rroof. By Proposition 1, U e m (t (2U \ c(X ))) if and only if there is some compact set C C c(X) with ^c(X) (c(X) \ C) x (c(X) \ C) C Int(U). 60

First we show that C m (t (2U\c )). Given

A £ c(X) and R C X x X with A^ C Int(R) we know A by Lemmas 15.1 and 15.4 and Proposition 15 that 2 is a closed compact subspace of c(X), and then we have

(c(X) \ 2A ) x (c(X) \ 2A ) is an open subset of i/(A,R).

It remains to show that A2 A IntV(A,R). Take B e c(X) with A C Int(B). Since Ag C Int(R) then

R n BxBel/|g. Take U e u with UnBxBCR, and take V e u with V O V C U and V[A] C B. We assert that H(V)[C] x H(V)[C] C l/(A,R) for each

C e 2A . Given D,E e H(V)[C] we have D C V[C] C

V O V[E] C U[E] and E C V[C] C V O V[D] C U[E].

D and E are both subsets of B since V[C] C V[A]

B, so D C R[E] and E C R[D],

Now we show that given U e M (r(2y|c )) there is some U(A,R) e sx with l/(A,R) C (J. Take a compact set C C c(X) with (c(X) \ C) x (c(X) \ C) C II and let A = UC. A e c(X) by Lemmas 15.2 and 15.3. Since

U e 2^|c^jq there is some U e u with H(U) l~) c(X) x c(X) C ' U. We note at this point that A^ C Int(U) and

C C 2A . We will be done if we show that V(A,U) C U.

Given (B,C) e t/(A,U) suppose B C A or CCA. Then

(B,C) e H(U) C U. Otherwise B and C are elements c(X) \ C so (B,C) e U. Q.E.D. In Chapter 2 we looked at how the cofine uniformity behaves with respect to taking images, subsets, products and sums. With the next two Propositions we can answer the same sort of question with respect to taking the hyperspace. Adapting the notation of Michael, let F^(X)

= {c{x}|x e X} and let (X) = {c{x,y} |x,y c X}.

Proposition 20. Given (X,y) and A C 2^ with

F^(X) C A, if 2^1^ is cofine then A \ c(X) is nowhere dense.

Proof. If 2 U\^ is cofine then so is U, by Proposition

7 and Lemma 15.6. We must show that c(A \ C(X)) has empty interior. Suppose there is an 0 e t ( 2 U) with

0 ^ 0 C c(A \ c(X)). There is some A e 0 fl A \ c(X) .

By Proposition 2, A must be a point of local compact­ ness of (A,2U|^). So there is an open set 0* with

A e 0* C 0 and a compact set C which is the closure, relative to A, of 0* n A.

We assert that the closures, relative to 2 X , of v" I-’- and C are the same. Certainly cC C c0*. On the other hand, if A* e c0* then any open neighborhood N of A‘f contains a point of 0*, and N n 0* contains a point of A since 0* C 0 C cA, and so N contains a point of 0* n A C C. 62

So c0* is compact since it is the closure of the compact set C. Then A is a point of local compactness Y of 2 , contradicting Proposition 17, since A £ c(X).

Q.E.D.

V Proposition 21. Given (X,c/) and A C 2 with

F^(X) C A, if 2t/j^ is cofine then (X,t/) is compact.

Proof. Without loss of generality, (X,f/) is not indiscrete. So take U e u with U £ X x X, and then

take V e u with V = and Vo V C U. Then for any x e X we have V[x] £ X (otherwise X x X =

V[x] x V[x] = V O V = U). Let = {c{x}| x e X} C A and letA^ = {c{x,y}|(x,y) e X x X \V} C A. Take an open symmetric Wet/ with W o W C V. Then H(W)[A^] n

f<2 = 0; for suppose (x,y) £ V and c.{x,y} e H(W)[A^]. Then x,y e W[c{z}] for some z, so W[x] n c{z} £ 0 and W[y] n c{z} £ 0. < Since W is open, z e W[x] n

W[y]. So (x,z) and (z,y) are in W and we reach

the contradiction that (x,y) e W O W C V.

Now by Proposition 2, one of the sets A^ or A2

must have compact closure. If it is A^ then (X,t/) is

compact, since by Lemma 15.6, A^ is closed and is

the T^-identification of X. So suppose A2 has compact

closure; we must show that this implies (X,t/) is compact.

It suffices to show that (X,t/) is complete, since if A 63 inherits the cofine uniformity then (X,£/) is cofine by Proposition 7 and Lemma 15.6, and so (X,t/) is totally bounded by Proposition 2.

Given a Cauchy net S : D -*■ X we choose for each d e D a point e X with Sd 4 V[S^]. Then let

S*:D -*■ ^ 2 be defined by S*d = c{Sd,Sd}. There must be a C e 2X which is a cluster point of S*. Take e u with and O C W. Take d' £ d . so that d^,d 2 £ d implies (Sd ,Sd ) e W ^ . Take d' 2 d such that S*d , e H(W1)[C]. Since Sd » e S*d , C W ^ C ] , there is an x e C with Sd i e W^[x]. Then for any d^ 2 d,

Sd* c W^[Sdi] C Wx O W1[x] C W[x]. Now if Sd* were also in W[x] we would have (S'd*»Sd*) e W[x] x W[x] C

W O W C V. So for each d* 2 d, Sd* t W[x].

We assert that S clusters to x. Since x e C C lim S*, if we are given 0 e r( u) with x e 0, and given a d'1 e D, there is a d* 2 d,d'' such that 0 l~l W[x] f“l

S*d* 4 0. Since 0 n W[x] is open, either Sd* or

Sd* is in O n W[x]. But S^d* 4 W[x]. So Sd* e 0. Q.E.D.

Corollary 21.1. If C A C 2X and A is closed, then 2^1 is cofine if and only if A is compact. A

Corollary 21.2. (2^,2^) is cofine if and only if it is compact. 64 X This result that (2 ,2 u ) must be compact in order to be cofine is fairly restrictive. We turn then to the investigation of when a hyperspace is locally compact and thus admits a cofine uniformity (necessarily different from the hyperuniformity when the space is not compact).

Proposition 22. Let (X,u) be an arbitrary space. The following statements are equivalent. X U 1. (2 ,2 ) is locally compact at the point X.

2. (X,t/) is uniformly locally compact, and there is a U e u with the property that for any V e u there is a compact set C C X with U[x] C V[x] whenever x e X \ C.

3. X is a disjoint union of sets {^a ! <* £ L} with the following properties:

a. Each Xa is compact.

b. U {X x X I a e L} e £/. a a 1 c. For any Vet/, there is a finite set F C L such that X x X C V whenever a e L \ F, i.e., the a a sets {Xa | a e L} are "almost all small." X u 4. (2 ,2 ) is uniformly locally compact.

Proof. 1 implies 2. Take U e u so that c(H(U)[X]) is compact in 2^. Take Vet/ with V o V O V C U,

V closed, and V = V Take W c u with W C V,

W open, and W = W -"*". Then we assert that V[x] is 65 compact for each x e X. Given a net S:D V[x] let

S*:D -v H(U)[X] be defined by S*d = (X \ W O V[x]) U c{Sd } e 2X . (X C U[S*d ] since W O V[x] C

W o V o v[sd] c u[sd].) Take B e c(H(U)[x]), a cluster point of S*. We assert that B n V[x] f 0. Take d e D so that S*d e

H(W)[B]. Sd e S*d C W[B] so we can take y e B n

W[Sd ]. Then y e W O V[x]. Suppose y £ V[x]. Then y e W o V[x] \ V[x] e r(u) and W o V[x] \ V[x] is disjoint from every S*d for d e D. So y £ Tim S*, but this contradicts y e B. So y e B n V[x].

We will show that S clusters to this y e B n V[x].

Given d e D and y c 0 e r(u) we can take d 1 2 d such that O' n W[y] n S*d t £ 0, since y e lim S*. Take z e O D W[y] n S*d i; since y e V[x] we have z e W O V[x] and so z e W o V[x] n S*d » = c{sd i}•

Then z e 0 n c {Sd* } so Sd t e 0. We have shown that every net into V[x] clusters in V[x], so V[x] is compact for every x, and so (X,t/) is uniformly locally compact.

We now change our choice of U e u so that c(H(U 0 U)[X]) is compact, U[x] is compact for each x e X, and U = U-^. Given Vet/, we will show that

B = {x e X| U[x] CX V[x]} has compact closure. Since a uniformly locally compact space is complete, it suffices 66

to show that B is totally bounded. Since shrinking V just makes B larger, we can assume without loss of generality that V is open and symmetric, and V C U.

Given a net S:D -*■ B, let S“ :D -*■ H(U O U)[X] be defined by S*d = X \ V[Sd] e 2X. (X C U O U[S*d ] since X = S*d U V[Sd] C S*d L) U[S^] and, since

U[Sd] \ V[Sd] ^ 0, Sd e U[Swd].) Take a cofinal map f:D' -*■ D so that S* o f is Cauchy. Take d' e D' such that d^,d2 2 d' implies (S* o f(d^),S* O f(d2 )) e H(V).

Then for any d^ a d' we have S O f(d^) e V[S O f(d')] for the following reason. If S o f(d^) i V[S o f(d')] then S o f(d^) e S* O f(d') C V[S* o f(d^)] and so

V[S O f(d^)] n S* O f(d^) ^ 0, a contradiction. Now

V[S O f(d')] C U[S O f(d')], which is compact. So the subnet S o f of S is eventually in a compact set, hence it has a Cauchy subnet, which in turn is a subnet of S.

2 implies 3 . Assume that U, in addition to the property that for any Vet/ there is a compact A C X with

U[x] C V[x] for x e X \ A, also is open and symmetric, and for any compact A C X the set U[A] has compact closure. Take Vet/ with V o V C U. There is a compact set A such that x i A implies U[x] C V[x].

Let B = c(U[A]); then B is compact. If x^ 0 B and

X 2 e U[x^] then X 2 i A, and so U[x2 ] C V[x 2 l* We 67 then have U O U[x^] = U{U[ x 2 ]| ^ 2 £ U[x^] } C U {V[X2 3 | x2 e U[xx]} = V o U[xx] C V o V[x1] C U[xx]. So if x £ B then U[x] n B = 0; this is because A (~l U[x] =

0, so A n U O U[x] = 0, so U[A] (“I U[x] = 0 since U = U_1, and finally B n U[x] = 0 since U is open. Let U' = B x B UUet/. We will show that U 1 is an equivalence relation on X. It suffices to show U' is transitive. Given (x,y), (y,z) 0 U', if x B then

(x,y) £ U, so y £ B and so (y3z) £ U. Then (x,z) e

U o U so zsUo U[x ] C U[x ] hence (x,z) £ U C U'.

If, on the other hand, x e B then y £ B, since other­ wise (x,y) e U so (y,x) £ U and so we would have x B. Similarly z e B. So (x,z) e B x B C U 1 .

Let {Xa| a e L} be the equivalence classes of U'.

Note that U 1 = U {1 X ^a x a X 1 I a e 1L}. It follows that each X a is clopen,v 7 since cX^ a C U'[X u aJ1 = X„, a and if x £ X a then x £ U'[x] L J C X a. Each X^ a is compact r since either X^a = B or X„, a = u U[x] J for some x and U[x] = c(U[x]) is compact. -1 Given Wei/ take Wq e u with Wq = Wq and

W0 ° W0 Cl W. There is a compact set A^ C X, where

U[x] C Wq [x ] whenever x £ A^. We may assume B C A^

Then {Xa | a e L } is an open cover of A^, so there is a finite F C L with A^ C L){Xa| a e F}. Then whenever a ]L F, we have X^ x Xa = U[x] x U[x] C 68

W Q [x] x WQ [x] C WQ o WQ C W.

3 implies 4. Let X = U { | a e L} satisfying property

3, and let U =u{Xa x X^| a e L} e u. We note that each 2Xot is compact. Given A e 2X, let A = H(U)[A]. ^ y Then A is just the set of A" e 2 ‘ such that A and

A* intersect the same equivalence classes of U.

We will show that A is compact. Given an ultra­ filter f on A we will show that F converges in A .

Let L 1 = {a e L| A (“1 X Q ^ 0} and for each a e. L 1 and each A* C A let f^ (A*) = {A* n X j A* e A}. Then let f = { f ( F) I F e f}. We claim that each F is an a 1 a 1 a ultrafilter on 2Xot.

0 i F since if A“ e A and a e L1 then A* a Xo * 0.

If F1>F2 e F ttien fa ( Fi) n fa ^F2^ “l fa^ F1 n F 2^ £ f . a If f ( F) C F c 2Xa with F e f, then let a a 7 P'f = {A* e A | A* n Xa e Fa}. F C F* so F* e f, and f ( F*) = F so F e f . a a a a Finally, if 2X<* = 81 U B2 , let B1* = {A* e A|

A* n Xa e B1 } and let B2* = {A* e A|A* n Xa e B2). Then A = B^* U ^2*, so one of B^* and B2" is an element ofF. 8^ = atld ^ = fa C^2*). So one of B 1 and 2 B is an element of f a . 69

So for each ct e L’ we can choose C e 2 a to a be a limit of Let C = U {Ca | a e L' }. We will show that F converges to C. Given Vet/ with

V C U, there is a finite set E C L' with X^ x Xa C

V whenever a e L* \ E. For each a e E we have

H(V)[C0] e Fa, so {A* e A |A* n Xa e H(V)[Ca]} e F.

The intersection of this finite collection of sets is

F” = {A"* e A | fa(A*) eH(V)[Ca] for each a e F.} which must therefore be an element of F. But F" C H(V)[C], since given F'f e F,f, if a e E then Xft n F'f e

H(V)[Ca] and if a e L \ E then X a H F* C

(xa x Xa)[Ca] Q V[Ca] and likewise Ca C V[Xa n F*].

So A* = U {A* n Xj a e L'} C U {V[Ca]| a e L ’} =

V[G] and likewise G C V[A*].

Lastly, we note that C e A , so F converges in A.

We have shown that H(U)[A] is compact for any A e 2 , y so 2 is uniformly locally compact.

4 implies 1 . Trivial. Q.E.D.

Example 3 . Consider R, the reals, with the usual metric d. Let X C R be the union of the collection of closed intervals { [n,n + n + g] I n = 1, 2, 3, . . . } .

Then X with the induced uniformity admits a locally compact hyperspace. For the relation when restricted to X is an equivalence relation, and its 70 equivalence classes are all compact. Furthermore, if

V, is given with e >0, take a whole number Cl y £

N > ^ . Then if n £ N we have

[n.n + x [n,n + C Vdje> i.e.j

'In diam[n,n + — r—o] < g . So X is seen to satisfy 7 n + 2 property 3 of Proposition 22.

In the spirit of Proposition 19, we will construct S a subbasis for M(T(2U)) in the case when (2X ,2U) is locally compact. Let X = U{Xa|a e L} satisfying property 3 of Proposition 22, and let U = U{X^ x X^J a e L} g u. For each K C L let A^ = {A e 2X | A H

X a ^ 0 if and only if a e K}. Each A^ is clopen since A^ = H(U)[A^], and compact as seen in the above proof that 3 implies 4 in Proposition 22. For K C L, a* e K, and R C X x X, let W(K,ct*,R) = (2X \ A R ) x

2X \ A ) U H ( ( U { X a x X | a* ^ a e K}) UR). Then let

= {W(K,a*,R)| a* g K C L and A^ C Int(R)}.

v u Proposition 23. If (2 ,2 ) is locally compact, thus admitting a cofine uniformity A/(r(2t/)), then flX* is a subbase for 2U)).

Proof. First we show each f-,'(K,a*,R) g m(t(2U)). Since X X A^ is closed and compact, and (2 \ A^) x (2 \ A^) C

W(K,a",R), it suffices, by Proposition 1, to show that

A2 X c lnt(w(K,a*,R)) whenever Ax Int(R). Since 71

X * is compact, R n X * x X * e tf|v *• Take U* e v a a a 1 Xa. ’ with U* fl Xa* x Xa* C R and U“ C U. Take V e 1/ with

V O V C U*. We will show that for A £ 2^, H(V)[A] x

H(V)[A] C W(K,a*R).

Given B,C e H(V)[A] we have B C V[A] C V O V[C] C

U*[C] and likewise C C U*[B]. If either B e A^ or C e then both are, since H(U*)[A^] C H(U)[A^] c

A^; if neither B nor C are in A^ then (B,C) e

W(K,a*,R). So we assume B,C z A^. B fl Xa* C U*[C] so we have B fl Xa * C U*[C (1 X a-vl, since if a 5^ a* then U*[Xa ] n Xft* = 0. Similarly C D Xa * C U*[B n Xa* ] . So (B n Xa*,C n Xa*) e H(R). Then B = U {B n Xa | a e K} C ( U { X a x Xa [C]| a £ K\. { a*}}) U

R[C] C ( ( u { X a x Xa | a* f a e ft}) U R)[C] and vice versa. So (B,C) e H(((J {X^ x X^ | a" ^ a e K} ) U R) C W(K,a*,R).

It remains to show that given U e M(r(2^)) there is some finite subset of w^ose intersection is con- X tained in U. Take a compact set C C 2 such that

(2^ \ C) x (2^ \ C) C U. Since the sets A^ form a disjoint open cover of 2 , there are sets , . . . ,

K C L such that C D A ^ 0 if and only if K = K. n ft J for some j from 1 to n. Now ti £ 2 , so take a

U* £ U with H(U*) C U and U‘v C U. Take a finite set F C L such that if a e L \ F then 72

X x X C Uv, and F D K. £ 0 for each j. We assert a a 7 3 that fl {W(K, a*,U") | a* e F fl K and K = for some j} c u.

Given (A,B) e W(Kj,a*,U") for each j from 1 to n and each a* e F n K ^ , if A,B £ C then (A,B) e d.

Suppose A c C; then A e for some j. J For each a* c K. n F, (A,B) e H ( ( U {Xrt x X I a* £ a e K. }) U U*). J OC Ot J So B e A, , since K- (1 F £ 0. Furthermore, for each TCj J a e Kj fl F we have (A (“I X a,B D Xa) e H(U“)• If> on the other hand, a e Kj \ F, then (A l”l X a,B fl X a) e

H(Xa x Xa) C So (A,B) e H(U*) C U. If instead we assume B e C, we can show (A,B) e (J in exactly the same way. Q.E.D. CHAPTER 5 CHARACTERIZATION OF HYPERSPACES

When is a given space unimorphic to some hyperspace?

We take a quasi-algebraic approach to this question, noting that a hyperspace is endowed with a binary operation which assigns to two sets their union. This operation turns out to have certain nice properties relative to the hyper­ uniformity. For instance:

Proposition 24. [Michael] On an arbitrary hyperspace X TJ (2 ,2 ) regard the union operation as a map U ! X x X -*■ X.

Then U : (X,y) x (X,i/) -► (X,i/). Proof. Given A,B e 2 and U e u we can show that if

A* e H(U)[A] and B* e H(U)[B] then A* U B* c H(U)[A U

B]. Thus: A* U B* C U[A] U U[B] and A U B ‘ <=

U[A*] U U[B*] = U [A* l) B*].

If *: X x X -*■ X is a binary operation on X, assigning to each pair (x^,X2 ) a point x^ * X£> we define a relation £ on X by saying that x^ £ X2 whenever X2 * x^ = X2 * bet A^ (or, in the absence of the possibility of confusion, just A) be the set

{a e X| V x e X, if x £ a then x = a} . For S C X

73 74 let = {x e X| x £ y for some y e S }. If u is a uniformity for X with *: (X,t/) x (X,u) ■> (X,ct) we call

(X,t/,*) a uniform groupoid (by analogy with topological

V TT group). Thus by Proposition 24, (2,2 , U) is a uniform groupoid.

Proposition 25. Let (X,t/,*) be a uniform groupoid. For there to be a space (Y,v) and a unimorphism g: (X,u) ■*

(2^, 2 V) such that for each xi »x 2 e ^ we have g(x^ * X2 ) - g(x^) U g(x2 ) it is necessary and sufficient that (X,t/,*) satisfy all of the following properties:

1. (X,u) is T 2. 2. *: (X,t/) x (X, £/) (X , £/) is uniformly open.

3. * is associative.

4. * is commutative. 5. x * x = x for each x e X.

6 . For x ^ » x 2 G if {x^ n Ax C {X2 then x^ £ X2* 7. For S C X, if S / 0 then there is a point x e X with S C {x and {x}^ D A^ C c(S^).

8 . For any U e u there is a Vet; such that for any a e A^ we have V[a]^ C U[a]. 9. For any U e u there is a Vet; such that for any x p X there is a y e U[x] with V[x] C {y}^ -

10. - For any U e v there is a Vet; such that for any x e X we have {y e X| x £ y e V[x]^} C U[x]. 75

The next six lemmas will be used to prove the sufficiency side of Proposition 25.

Lemma 25.1. If (X,*) satisfies properties 3, 4, and 5 of Proposition 25 then £ is a proper partial order on X.

Proof. If xi s x 2 ^ then x^ - x^ * X 2 = x^ *

(X2 * x^) = (X3 * X2 ) * x^ = Xg ic x^ and so x^ £ x^.

If g X2 and X2 £ x^ then X2 - X 2 * x^ = x^ * X 2 = x^. Finally x s x since x * x = x.

Lemma 25.2. If (X,*) satisfies properties 3, 4, and 5 then x^ * X2 is always the least upper bound of x^ and

X2 with respect to the partial order S.

Proof. We note that (x^ * X2 ) * X2 = x^ * (X2 * X2 ) = x^ * X2 so X2 £ x^ * X2 « Then since x^ * X2 - X2 * x^ we have x^ £ x^ * X2 * Suppose x^,x2 * X* Then since y * (x^ * X2 ) = (y * x^) * X2 = y * X2 = y we have x^ * X2 £ y.

Lemma 25.3. Let (X,*) satisfy properties 4-6. Then for any x e X we have { x } ^ (1 A^ ^ 0.

Proof. Suppose {x} ^ n A = 0. Then for any y e X we have {x }£ fl A C {y } ^ so x £ y by property 6 . In particular, if y £ x that implies y = x. This shows that x e A. But x £ x so x e {x}^ D A, a contradic­ tion .

> 76

Lemma 25.4. If (X,t/,*) is a uniform groupoid satisfying property 1 then for any x e X, {x}^ is closed.

Proof. Let R^ = *""^[x] = {(x^,X2 ) G X x Xj x^ * X2 = x}.

Since {x} is closed and * is continuous then R is a closed relation. Then {x}^ = {y e X| y £ x} = {y e X| x = x * y} = {y e X| (x,y) £ Rx } = R x[x].

Lemma 25.2. Let (X,t/,*) be a uniform groupoid satisfying properties 1 and 3-7. Suppose 0 ^ S C X. Then there exists a unique least upper bound of S, denoted by lub S.

Furthermore, {lubS}^ n A^ = A^ fl c(S^).

Proof. By property 7 there is an x s X which is an upper bound for S and satisfies {x }^ D A C c(S^). Given y e X with S c {y } ^, we have {x}^ fl A C c(S^) C c({y}^) - {y}£. So x s y by property 6 . Then x is a least upper bound for S, and of course least upper bounds are unique when the partial order is proper. Since c(S^) C c({x}£) = {x}^ we have {x}^ fl A = A (“I c(S^).

Lemma 25.6. Let (X,t/,*) be a uniform groupoid satisfying properties 3, 4, 5, and 10. Then for any U e u there is a Vet/ where whenever x £ V[y]^ and y £ V[x]^ then

(x ,y) £ U. Proof. Take Uq e u with o Uq C U. Take £ u so that when x £ z and z e U^[x]^ then z e UQ [x].

Take Vet/ such that V[x] * x C U^[x]. Suppose x £ y* e V[y] and y £ x' e V[x]. Then y* * y e U^[y]

and x' * x e U^[x]. By Lemmas 25.1 and 25.2, we have

x,y £ y' * y and x,y £ x ’ * x, and therefore x * y £

y* * y and x * y £ x' * x. So x * y e U^[x]^ (“1 U^[y] * and of course x,y £ x * y, so x * y £ Uq [x] n u0[y].

Finally, y e Uq "^[x * y] C 0 U tx].

Proof of sufficiency in Proposition 25. Assume (X, y,*)

is a uniform groupoid satisfying properties 1-10. For

any x e X let g(x) = {x}^ fl A. g(x) e 2^ by Lemmas

25.3 and 25.4. We assert that g(x^ * X2 ) = g(x^) U g(x 2 ). By Lemmas 25.2 and 25.5 we have g(x^ * X2 ) =

g(lub {x^,X2 }) = A fl c( {xlsX 2 }^) = A fl c( { x ^ \j {x2 } * )

A n (cUx^5) U c({x2}S) = A n ({xjJ6 u {x2}s) ■ g(x1) U g(x2). We must show g: (X,y) -*■ (2^,2U \A) is a unimorphism.

By property 6, g is one-to-one since if {xil^ n A =

{x2 }S fl A then x^ £ X 2 and X 2 £ x^, and so x^ = X2 * A We assert that g is onto. Given B e 2 let x = lub B.

Then g(x) = {x}^ fl A = A D c(B^) = A (“I cB = B. We assert that g is uniformly continuous. Given a

symmetric U e y, take V e u such that for any. a e A,

V[a]S C U[a]. Take a symmetric W e V such that for

any X^,X2 e X, W[x^ * X 2 ] C. X * V[x 2 ]. We assert

that W C (g x g)"^[H(U)]. Given (x,y)e W we show

g(x) C U[g(y)]. Given a e g(x), x * a = x so 78 y e W[x] C X * V[a], and so y = y 1 * y'' for some y' e X and y ' 1 e V[a]. By Lemma 25.3, there is some a* e {y' 1 }S D A. Then a* £ y and a* e V[a] ^ CU[a] and so a e U[a*] C U[g(y)]. The proof that g(y) C

U[g(x)] works the same way. So (g(x),g(y)) e H(U).

Finally we show g is uniformly open. Given U e t/, by Lemma 25.6 there is a V £ u with (x,y) £ U whenever x £ V[y]* and y e V [ x ] By property 9 there is a

W £ u such that for any x £ X there is an x ’ £ V[x] with W[x] C {x1 }S. Since * is uniformly continuous we can take Wq e u with Wq[x] * x' C W[x'] whenever x £ x'. We assert H(Wq n A x A) C (g x g)[U]. Given

(g(x),g(y)) £ H(Wq D A x A) we must show (x,y) £ U.

For a^ e g(x) there is an s g(y) with (3 2 ,3 ^) e

Wq. We have a 2 ^ y so a^ * y e w o [a 2 3 * Y c W[y] and therefore a^ e W[y]We have shown g(x) C W[y]S and an analogous argument gives g(y) C Wfx]Take x' £ V[x] and y' e V[y] with W[x] C {x' and

W[y] C {y1 }S. Then g(x) Cl {y1 }s and g(y) C {x‘ }? hence x S y 1 and y £ x' by property 6. So x £ V[y] and y £ V[x]and we conclude (x,y) £ U. Q.E.D.

Lemma 25.7. Given (X, t/,*) and (Y,tO and a unimorphism g: (X,£/) -► (2Y ,2U), if g(x1 * x2) = g(x1) U g(x2) for each x ^ > x 2 e ^ then: 79

a) x^ £ X2 if and only if g(x^) c g(x2 )> and

b) g[Ax ] = {c{y]| y e Y}.

Proof. a) x^ S x2 iff x2 = x2 * x.^ iff g(x2) = g(x2 * x^ iff g(x2) = g(x2) U g(x1) iff g(x1) C g(x2). b) If a e A suppose 0 8(a)* Take x e X with g(x) = c{y^}. • g(x) C g(a) so x £ a, and then x = a so c{y1 } = g(x) = g(a) = c,{ylty2 }. So y2 e °^yl ^ and we have shown g(a) = c{y^}. Conversely, if y £ Y take x e X with g(x) = c{y}. If x^ £ x then g(x^) C c{y}. Take y^ e g(x^). Suppose y £ g(x^). Then since

(Y,v) is regular, there is some neighborhood of g(x^) which doesn't contain y, contradicting y^ e c{y}. So y e g(x^) and hence g(x^) = c{y}. So x^ = x, and this shows that x e A.

Proof of necessity in Proposition 25. In view of Lemma

25.7, we can make the proof go more smoothly by identifying

(X,t/) with (2Y,2V), * with U, £ with C , and A with

{c{y } | y e Y } . We must show that (2Y ,2V , U) satisfies properties 1-10. (2 Y ,2 V ) is necessarily T2« We prove property 2 by showing that for U e v there is a V e v such that for

A,B £ 2Y we have H(V)[A L) B] C {C U D| C e H(U)[A] and D e H(U)[B] }. Choose V so that V O V C U and

V = V"1. Given E e H(V)[A U B], let C = c(E n V[A]) 80 and let D = c(E n V[B]). E C V[A U B] = V[A] U V[B] so E = C (J D. C C c(V[A]) C U[A]; on the other hand

A C V[E], so given x e A there is a point x^ e E with x e V[x^]. But then x^ e V[x] C C, so we have shown that A C V[C] C U[C] and hence (A,C) e H(U). Similarly

To prove 6, suppose {c{.x}| c{x} C B ^ } C {c{x}| c{x} C

B 2 }. Then clearly B^ C B 2 * Property 7: Suppose 0 C C 2^. Let B = c(UC); then if C e C we have C C B. We must also show that if c{x} C B then c{x} e c{D e 2^| D C C for some

C e C}. But x e c(UC), so given U z v with U = U ^ we have U[x] fl (UC) ^ 0. Take C e C with U[x] P

C ^ 0 and then take x* e U[x] p C. Let D = c{x“ }. We have DCCeC, DCUo U[c{x}] and c{x} C U o

U[D], so H(U O U)[c {x}] P {D c 2Y | D C C e C } ^ 0. Since this is true for any symmetric U e v we must have c(x) e c{D e 2Y | D CCeC}. Property 8: given U z 2V take an open symmetric

U z v with H(U o U) C U. Then for any c{x} e 2^, if B e H(U)[c{x}] and C C B then C C U[c{x}] and so

U[C] C c{x} ^ 0. Then x e U[C] and so c{x} C U O U[C].

So C e H(U o U) [c {x}] C U [c{x}].

Property 9: given U e 2V take U z v with H(U o U) C U. Then for any A z 2^, let B = c(U[A])e2^. 81

We have B C U O U[A] and A C B, so B e H(U O U)[A] C

U[A]. If C e H(U)[A] then C C U[A] C B. v Property 10: given U e 2 take U e V with H(U) C U. For any A e 2Y , if A C B C G e H(U)[A]

then B C U[A] and A C B, and so B e H(U)[A] CU[ a ]. Q . E. D .

Part of the significance of a result like Proposition

25 is that we can put our investigation of hyperspaces

into a more abstract framework. We may consider various

properties of hyperspaces to see if any of them depend on

some but not all of properties 1-10 in Proposition 25.

Proposition 26. Let (X, u, *) satisfy properties 1-6

of Proposition 25. Then A^ is closed.

Proof. Suppose that {x} ^ Cl A = {a} , and we will show x e A. Given y S x *we must have {y}^ D A = {a},

since {y} ^ fl A C { x} ^ fl A and {y}^ fl A y4 0 by

Lemma 25.3. So {x}£ fl A C {y}^ and therefore x £ y

by property 6. So x = y, and that means x e A. So if x 4 A then there are at least two points

a^,a2 £ {x} * fl A. Take U e u such that U[a^] fl

U[a 2 l = 0. Take Vet; such that V[x] C (X * U[a^]) D

(X * U[a2]). Then for any x* e V[x] there must be an

a* e U[a^] and an a2 e U[a23 with a ^ > a 2 5 x** So x* A, since aj ^ a^. This means that V[x] fl A

= 0. Q.E.D. 82

Knowing that is closed when (X, £/,*) satisfies

1-6, we know that under the same conditions would inherit from X any of a certain class of properties.

If (X,t/,*) satisfies 1-6 and is compact, locally compact, complete, normal, paracompact, lindelof, or countably compact, then A^ would be compact, locally compact, complete, normal, paracompact, lindelof, or countably compact, respectively.

If 5 is a proper partial order on X, and S: D X is a net, let lim sup S be the greatest lower bound of the set {lub S[d,co)| d e D} whenever it exists. Then in a hyperspace lim sup S corresponds to our previously defined lim S.

Proposition, 27. For an arbitrary space (X,tf) regard X u X (2 ,2 , L) ) as a uniform groupoid. Given a net S: D -*■ 2 , either lim S = 0 or lim S = lim sup S.

Proof. In a hyperspace, lub S[d,°°) must be the smallest closed set containing each set f°r d* S d, i.e., lub S[d,°°) = c( U S[d,“ )) . But US[d,«>) = as defined in Proposition 14.d. Then by the same Proposition, lim S = cluster(Fg) = ClfcF^I d e D} = fl {lub S[d,»)| d e D} which, if it is non-empty, must be lim sup S.

Q • E • D •

Proposition 28. Let (X, ^,*) satisfy properties 1-7. Sup­ pose S: D X is a net which clusters to x e X. Then 83

lim sup S exists and x S lim sup S.

Proof. Given a e {x}^ n A and d e D we assert

a e c(S[d,°°)^). For U e u, there is a Vet/ such that

V[x] C X * U[a]. Then there is a d 1 £ d with S^, e

V[x], and so there is an x' e U[a] with x' £ .

Hence S[d,«>)^ n U[a] ^ 0.

By Lemma 25.5, A n c(S[d,°°)^) = {lub S[d,°°)}^ fl A.

So for each d e D, {x}S D A C {lub s[d,<»)}^ and so x £ lub S[d,°°) by property 6. Hence x is a lower bound

for {lub S[d,°°)| d e D}. If L is the set of such lower bounds, then L has a least upper bound by Lemma 25.5.

We then have x 5 lub L. We also have lub L = lim sup S

since for each d e D, lub S[d,°°) is an upper bound for L so lub L £ lub S[d,°°). Q.E.D.

It is interesting to note that Propositions 26 and 28 do not use the full strength of property 2. The proofs can be done assuming only that * is an open mapping, not necessarily uniformly open.

Proposition 29. Suppose (X,£/, *) is a compact uniform groupoid satisfying properties 1-8. Then (X, £/,*) also

satisfies 9 and 10.

Proof. Ir. the proof of sufficiency in Proposition 25, we used properties 1-8 to show that there is a one-to-one,

onto, uniformly continuous map g: (X,t/) (2^, 2 ^^). 84 A Then if X is compact and 2 is T 2 , g must be a

A *1 homeomorphism. 2 is also compact, and this forces g” to be uniformly continuous. So g is a unimorphism. Then by the necessity side of Proposition 25, (X,£/,*) must satisfy properties 9 and 10. Q.E.D.

Finally, here is a simple application of the ideas of this chapter to local compactness.

Proposition 30. Let (X, t/,*) be a uniform groupoid satisfying property 2. Then if x^ and X2 are points of local compactness of (X,t/) so is x^ * X 2 *

Proof. Let and N 2 be compact neighborhoods of x^ and X2 respectively. Then since * is both open and continuous, ^ is a compact neighborhood of x^ -,v X 2 • Q.E.D. LIST OF REFERENCES

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