Semi-Uniform Spaces and Hyperspace Constructions

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This dissertation has been microfilmed exactly as received 70-6815

KLEIN, Albert Jonathan, 1944- SEMI-UNIFORM SPACES AND HYPERSPACE CONSTRUCTIONS.

The Ohio State University, Ph.D., 1969 Mathematics

University Microfilms, Inc., Ann Arbor, Michigan

g) Copyright by

Albert Jonathan Klein

1970 SEME-UNIFORM SPACES

AND

HYPERSPACE CONSTRUCTIONS

DISSERTATION

Presented in Partial Fulliment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Albert Jonathan Klein, B.Sc., M.Sc.

The Ohio State University

1969

Approved by

(fVW[OtUL ofj& A A A Adviser Department of Mathematics ACKNOWLEDGMENTS

I would like to thank Professor Norman Levine for his many valuable suggestions and my wife, Susan, for her accurate typing and quiet forbearance.

i i VITA

November 16, I9l4.il- • • • Bom - Dayton, Ohio

1966 ...... B.Sc., The Ohio State University, Columbus, Ohio

1966-1968 ...... Teaching Assistant, Department of Mathematics, The Ohio State University, Columbus, Ohio

196 7 ...... M.Sc., The Ohio State University, Columbus, Ohio

I 968- I 9 6 9 ...... NDEA Fellow, The Ohio State University, Columbus, Ohio

FIELDS OF STUDY

Studies in Point-set and Uniform Topology. Professor Norman Levine

Studies in Real Analysis. Professor Bogdan Bajsanski

Studies in Algebra. Professor Arnold Ross

i i i TABLE OF CONTENTS

ACKNOWLEDGMENTS ......

VITA ......

NOTATION......

INTRODUCTION...... „ ......

Chapter

I . SEMI-UNIFORM SPACES......

General Facts Semi-Uniform Spaces with Property K

I I . THE HYPERSPACE OF A SEMI-UNIFORM SPACE ....

Definition and General Facts HK Spaces Topological Properties

H I . THE HYPERSPACE OF A TOPOLOGICAL SPACE . . . . .

Definition and General Facts Comparisons with the Finite Topology HK Spaces Weak Separation Properties Completely Regular Spaces Compact Spaces Locally Compact Spaces

IV . PSEUDO-METRIZABILITY AND THE HYPERSPACE OF A TOPOLOGICAL SPACE ......

The H ausdorff Pseudo-M etric Pseudo-M etrizability of Uniform Topologies Pseudo-Metrizability of Fine Uniformities Pseudo-Metrizability of the Hyperspace

V. SOME QUESTIONS ......

BIBLIOGRAPHY NOTATION

The following notations w ill be used without comment. Here

X and Y are sets,

1) A s = I C x,x)jx£X ^is the diagonal of X, (When only one set is involved, A w ill denote- the diagonal.)

2) X = ^a | a s X^' is the hyperspace of X.

3) For SSX*X, S'1- £(x,y)| (y,x)£ s],

k) For S£X*X and AS.X, s£a] = £x}(a,x) <£ S for some aeA^.

5) Let f:X-»Y. f*f:X*X-*Y*Y by fxf(x]_,x2)= (f(xx),f(xg)).

6) Let f:X-+Y, let A«=X and let BSY. f/X} = £f(a)|aeA ^ and f~ ‘j¥]=.[x|f(x) 6 B^.

7) Let A, B X. (°rCA) is the complement of A in X.

The notation A-B=A/\

8) Let ^ be a topology for X end let A£X. The closure of A is cA and the interior of A is Int A.

9) Let *¥ be a topology for X and let ^ be a topology for Y.

Let fsX—>Y. f :(X,1^) — Y/%) w ill mean that f is continuous.

10) Let be a topology for X and let Yi. be a topology for Y.

means that (Y,^) is a subspace of (X ,^).

11) Let Y be a topology for X and let A£X. A AY will denote the relative topology on A.

12) Let fsX-^Y and let'Y be a topology for Y. is the weak topology for X induced by f and Y , v 13) Let f:X—5-Y be onto and let be a topology for X. £(?¥) the quotient topology for Y induced by f and ^ .

Ill) Let d be a pseudo-metric for X. For xeX,

Sr(x)»fyjd(x,y) 0, V£ (or Vd^£) is £(x,y) jd(x,y)<,

uniformity with base ) £ >o}.

vi INTRODUCTION

Attempts to induce natural topological and uniform structures on

the hyperspace (power set) of a set began with Hausdorff’s metric on

the closed, bounded subsets of a metric space. In the case of a uniform space the Hausdorff metric had a natural generalization which has developed into the standard hyperspace of a uniform space. (See

Caufield [ 3]•) However, in the case of a topological space, there was no natural generalization. The finite or Vietoris topology (for a

comprehensive presentation see Michael [ 13]) is one acceptable generalization since, on the closed subsets of a compact metric space, it agrees with the topology induced by the Hausdorff metric. There

are, however, other possible lines of generalization, one of which was

the starting point for this paper.

The basic idea is the following: given a topological space, apply the standard uniform construction to the neighborhoods of the

diagonal and derive a hyperspace topology as if the resulting hyper

space structure were actually a uniformity. This can be viewed as

combining the modern formulation of the Hausdorff approach to the hyperspace with Davis1 idea [6j of studying "uniform-like” structures for arbitrary topological spaces. In this paper the "uniform-like"

structures which occur are the semi-uniform spaces (loosely, uniform

spaces without the triangle axiom; see Cech [l;]). The paper has four parts: a study of serai-uniform spaces

(Chapter I), the construction of the hyperspace of a semi-uniform space (Chapter II), a study of some aspects of the hyperspace topology described above (Chapters III and IV),and a list of un answered questions and conjectures which have good potential for further research (Chapter V).

The following two points are of special interest. First of all, semi-uniform spaces with property K are introduced (section 2 of

Chapter I). In these spaces, the serai-uniformity is closely connected to the topology which it induces. For example, each entourage sectioned at any point yields a neighborhood of the point. Semi uniform spaces with property K are also closely connected to stable topological spaces (see Friedman and Metzler [.71, Davis [6] and

Naimpolly [lit]) • Secondly, there are several good theorems on pseudo-metrizability of the hyperspace and pseudo-metrizability of fine uniformities (Chapter IV). CHAPTER I

SEMI-UNIFORM SPACES

I.. 1. General Facts About Semi-Uniform Spaces.

I. 1.1 Definition. Let X be a set. A. non-empty family is called a semi-uniformity for X if and only if

a) A £ U for every U ettj

b) if U£V£ X*X and Ue'U , then V tU ;

c) if U and V e. U , then UA V eU ;

d) i f Uall , th en U~'elL „

I f il is a semi-uniformity for X, the pair (X/ti) will be called a semi-uniform space.

There are many examples of semi-uniform spaces; in particular, every uniform spa.ce is a semi-uniform space. (Clearly, a semi uniform space need not be a uniform space; many examples of this phenomenon w ill appear.)

The following theorem yields a class of examples of primary interest in this paper. •

I. 1.2 Theorem. Let (X,T) be a topological space and let be the neighborhoods of A in the product topology. Then (X,is a semi-uniform space.

Proofs since X*XeiProperties a, b and c above hold for the neighborhoods of any set. If U £ (X) and xeX, (x,x)£ O-jx O^s U fo r some 0^, . Hence,

(x,x)e OgXO^lT* and so IT's V.dt)»

From this point on, given a topological space (X ,^, °IX(1X) w ill always denote the semi-uniformity just described.

I. 1.3 Definition. Let (X,T) be a semi-uniform space and «3 and

(2> subsets of % .

a) is a subbase for ii if and only if

% = {u|/5 Si£U€X*X for some S-^...,S n £ $ } .

b) S is a base for "2t if and only if

V ~ ■[ u] B£ US X*X fo r some B e 0 j .

The next two lemmas w ill be used frequently to construct exam ples.

I. l.ii Lemma. Let X be a set. Let 2 — X*X. Then there exists a unique semi-uniformity % with subbase J? if and only if

a) A £ S f o r ev ery S £ S j

b) f o r ev ery S , 3 S^, « ..,S n £ w ith A S ^ S -1 .

Proof; The necessity of a and b is clear from the definitions of semi-uniformity and subbase. We show that a and b are sufficient. Let %= £U | A Us X*X for some S i,.. ,,Sn £ 2.J»

$ im p lies U^r C le a rly , il contains supersets of its members and is closed under finite intersections. Property a insures that

AS U for every DeU . Property b insures that S“ *£ i(. whenever

S 6% . Thus, since (A S.) = A S.”', U~,elL whenever U efy. , Thus, 1 Xsi i 11 is a semi-uniformity with subbase J . The uniqueness of follows from the definition of subbase. It is interesting to note that Kelley f10 J does not give a characterization of subbase for a uniform space.

I. 1.5 Lemma. Let X be a set. Let (&£X«X. Then there exists a unique semi-uniformity °U- with.base if and only if the following hold:

a) A £ B for every B c (S

b) for every B £. Q> , 3 B> £ /Swith Bs£ B

c) if B^ and B^e (5 ,3 B^£ CB> with B^z. B-jfi Bg.

Proof: The necessity of a, b and c-follows immediately from the definitions of semi-uniformity and base. That a, b and c are sufficient is shown as follows: Properties a and b together with the preceding lemma guarantee the existence of a unique semi-uniformity xd.th subbase(8 . Property c shows that <9 is in fact a base for1^ .

The next few definitions and theorems consist of standard uniform properties generalized to the semi-uniform case. The proofs are omitted, being exactly the same as in the case of uniform spaces.

(See Kelley [10]).

I. 1.6 Theorem. Let (XjlQ be a semi-uniform space. Then V. has a symmetric base.

I. 1.7 Theorem. Let (XfU.) be a semi-’uniform space and let Y£X

Let YxY/lW denote JlxYftuju c tt]. Then YxY A'U. is a semi-uniformity fo r Y.

I. 1.8 Definition. Let (Y,V) and (X/U) be semi-uniform spaces.

(Y,V) is a subspace of (X/U) if and only if Y^X and “V = YxY /)V.o

The notation (Y,V)£ (X,%) w ill mean that (YyY) is a subspace of (X/U)

I, 1.9 Definition. Let (XfU.) and (Y,*V) be semi-uniform spaces 6

and let f:X-*Y.

a) f is uniformly continuous if and only if Vc V implies

th at (fx'f)—1 [vj £°U . The n o ta tio n f : (X/U)-*(Y,V) w ill

mean that f is uniformly continuous.

b) f is a unimorphism if and only if f is 1-1, onto,

uniformly continuous and U cV. implies that fxf£u]cV.

I. 1.10 Theorem. Let (X3%) and (Y,V) be semi-uniform spaces;

let fsX-^Y. Then f : (X,^)-^(Y,V) if and only if, for every V£‘V,

3 such that (f*f)[u]£V.

I. 1.11 Theorem. Let (Xfii) and (Y,^) be semi-uniform spaces;

let f : (X$)-*(Y/V) b® I"-*- an<^ onto. Then f is a unimorphism if and

' only if f"1 : (Y,V)-Kx;U).

I. 1.12 Theorem. Let (X/U) be a semi-uniform space. Let

^(^) = [o^x|x£0^3 V etL with U£x]£ 0 . Then (X^CU)) is a

topological space.

From this point on, given a semi-uniform space (X/lX)y (X, ‘TX/1X))

w ill denote the topological space just constructed.

I. 1.13 Theorem. Let (X,iL) be a semi-uniform space. Then

0 £ *T(U) if and only if xeO 3 a symmetric U c'U such that

U fx ]£ 0 .

I. l.lli Theorem. Let X be a set with semi-uniformities % and

V. I f t y s V , th en (V ).

I. l.l£ Theorem. Let (X/U) and (Y,“V) be semi-uniform spaces;

l e t f : (X/U)-*( Y/V ). Then f : (X, ) )->{ Y, y;(V)). In addition, if

f is a unimorphism, f is also a homeomorphism.

Two standard uniform space theorems which fa il in the semi- uniform case are worth noting at once;

a) (Y/VOS (X#A) need not imply (Y,-T(V))^(X,''T(20).

b ) U ciL and xe.X need not imply that U[x3 is a

neighborhood of x.

Examples and a fu ll discussion of these phenomena w ill appear in

section two.

The next example shows that there need not be containment relations between a semi-uniformity H and ^ (U-)) « In particular, in the example, an entourage U^W-is not a neighborhood of A even

though UfxJ is a neighborhood of x for every x£X.

I. I«l6 Example. Let R be^the reals, let

X=-^(x,y)|-|x^y—2x and x> 0 ^ and l e t Y= |'(x ,y )J 2x^y

Let B= R*{o^Uf 0p

the y-axis and the points "between" the lines y = -|-x and y=2x).

Clearly, A £ B and B is not a neighborhood of (0,0) in the

usual topology on R^. Since B[0] = R, B[x3=|'yJ §x£- yf:2x^ if

x>0, and B[XI = ^yj2x^y£|'xj' if x<0, BfxJ is a neighborhood of

x for every x£R.

B = B“> is shown as follows: Let (x,y)cB.

If x=0 or y= 0, (y,x)£B. If (x,y)eX, then x> 0 and |x^y^2x which yields y> 0 and -|yt£x-£2y. Thus (y,x)e X^B. Similarly, if

(x,y)eY, (y,x)£Y.

Now l e t 'I' (x ,y )| |x -y |— € | for each C> 0 and l e t

B jk^o}-. Using A^B and B=B-1, it is easy to see that

& satisfies the requirements for a base given in Lemma 1.1.5* Thus

l e t V. be the semi-uniformity with base (8 . Let be the usual topology for R. Since j £ > Oj form a base for the usual uniformity on R, *¥ £ Let xcO g‘cT(^).

Then 3 C>0 such that (V6 H B)[x]£0. Since (VGn B)£x]=r V^x]/} B/>] and Bfx] is a ^-neighborhood of x, (Ve /I B)[x] is ^-neighborhood of x. Thus T = m > and so ‘^(■rc(‘^))= %(^)»

Since B is not a neighborhood of A , B

U. s.s follows: Let Sn=^(x,y)| jx-y|<^ and x,ya[-n,nj^.

Let v==^ S n. Clearly, V£2/(^). Suppose B<=.V f o r some

0 < £ < 1. Pick n positive such that -fc < £ and pick x> n. Let y=x-*£ . Then •|x:£:ys=2x so that (x,y)£B. Also (x,y)eV£ and so

(x ,y )£ V. Hence, (x,y)£Sjq- f o r some N*^.n-»-l and so )x -y } ^ -^ -< £ .

Contradiction.

The next theorem leads naturally to the definition of stable topologies. (See Friedman and Metzler [7]j also Davis [6],)

I. 1.17 Theorem. Let (X,1^) be a topological space. Then

Proof; This holds since, for each x£X and Us^UCT), U[x] is a

“^-neighborhood of x.

I. 1.18 Corollary. Let (X,“U) be a semi-uniform space and let lL* = U('r(U )). Then

P ro o f; Take * T = ^(& ) in 1 .1 .1 7 .

I. 1.19 Definition. Let (X,^) be a topological space. (X/t') is stable if and only if lL{%))—*¥ .

The next theorem contains several interesting characterizations of stable topological spaces. The theorem is in part due to Friedman and Metzler (a<ê belowj [7])* in part to Davis (bêj [6j), and in part to Levine (c<ê). I give a proof here for the sake of

convenience.

I, 1.20 Theorem. Let (X/^T) be a topological space. Then

the following are equivalent:

a) (X,^ is stable

b) for every xeX, c{x}= r\ -[UL'xjju £

c) f o r ev ery x,y<5"X, y e c{xj=^x

d) for every xeX, c^x^ — f] fojx eO e.1^ 1

e) for every xcOe'T, cfxj^O

P ro o f: (a=^b) L et ye cfxl and Let V<£ be

symmetric with V£U. VfyQ is a neighborhood of y and so xeYCyl.

Thus y e V£xJ £U[x]. Now let y £ f\ [uCxjj U£ If y e 0 e fT t

3 U £ tHC£) with U symmetric and U[yj£0. Since y£U[x], x £-Ufyj£E 0.

Thus y e c^xj.

(b^c) Let x,y£X with y& c{x}. Let U £ Vi*?)* Choose with V symmetric and V£U. Since V[yJ is a neighborhood of y,

V£y]£U[y]. Thus x £ A{ufxl|u£ 2Z(^)l=cfy?.

(c=?d) Let yecifx? and let xe O e^. If y^CO, cfy^C o which is impossible since xecfyj. Thus yeO. Let y£/^ojxe 0 e

Then xecfy] and so yec£x^.

(d=^e) Clear.

(e=j>a) By 1 .1 .1 7 , i t i s n ecessary to show on ly - ‘ti t l d ’Z)) •

Let xeO g'T. Let U=-0*0 U £c[x$ x (?c{xj. Then, since c{x^S0,

U £41(2 ). Clearly, Upx] S 0.

I. 1.21 Definition. Let (X/t) be a topological space. (X,^) is semi-uniformizable if and only if 3 a semi-uniformity ‘U. for X with 10

I. 1.22 Theorem. Every stable topological space is semi- uniformizable.

Proof: This is immediate from the definition and 1.1.2.

The next two examples show that the converse of 1.1.22 is false and that not every topological space is semi-uniformizable. A characterization of semi-uniformizable spaces appears later in this s e c tio n .

I. 1.23 Lemma. Let (X/T) be a topological space. For

■x.e.Q&'Y, let S ^ 0= 0x0 U C2£x5 and let J5 = -^Sx^q|x£0<£

Then a) is a subbase for a semi-uniformity U.

Proof: Each Sx^q contains A since x eO. Clearly, each Sx^q is symmetric. Thus, b y l.l.lj, i is a subbase for some semi uniformity U . L et x eO Then Sx q [x ] = 0. Thus 0 £ ^(H) and

L e v i t t ) - I. 1.2li Lemma. Let (X/7) be a topological space. Let xl,..*,xn£ X and xi€®i each i. Let Sv. n. be as in 1.1.23. Let x

a) /~\ S„ Q [x]= 0-F where and FS{x->, .. .,x ^ /t x i } i b) x ^ F if and only if x^O.^.

Proof: For each i, 0^ i f x = x ^

X i f x £ 0. and x^ix. SX i,0i [ ^ x x i f x ^ 0j_

Let F be the set of x^ such that S^q.LX] ~ 9 (^his yields "b11 im m ed iately .) Let 0 — A q^LÎ | Then it is clear that I. 1.25 Example« Let X be infinite and let A be a proper infinite subset of X. Let or A ô}. For any x£ A, c(£x})-=X. Then xt'A e'T but c(fx})Â. Thus (X/T) is not stable.

Now l e t U. be the semi-uniformity constructed in 1.1.23. It is sufficient to show that <> Let x eO f^ ). There exist

X p...,xn<= X and Op ...,0 n£ ^ such that/1 SXijOiLxJ^°* By I.1.2I;

3. 0* £ 15 such that 0*-F£0 where x^£ F<=^? x

F is finite, there exists ye A /0. Then there are m x and °l s u c h that /3 ^jO iTyJ-0* fiy i «i »^ 3 O’c such that yeO’-F'SlO where y^e F'^y^O |. Since ye A and each OjN£jzf, clearly F' = ^. Since O1^ ^ , A<=.O's.0. Thus Oe1/.

I. 1.26 Example. Let X — {a,b}; let {V, fa^,x|. (X/£) is not semi-uniformizable: Deny. Then there exists a semi-uniformity

*U such that :u) = T - Let V = /l{ u } u e %/j . V is symmetric and Ve2l.

Since {a'le'Y, V[a] = f a j , Since j[b3 4 ^ t V[b]=^a,b}. But then

V[a3 = J a ,b J • C o n tra d ic tio n .

I. 1.27 Definition. Let (X/ti) be a semi-uniform space. For each Ae X, le t k(A)-= £xju [x]/lA^=^ Vu clV.'}, The operator k will be called the weak closure associated with °li .

From this point on, given a semi-uniform space (X,^0, "k" w ill denote the weak closure associated with and "c" will denote the clo su re in th e to p ology 'VC 11)* As later examples will show, these operators may be different.

I. 1.28 Theorem. Let (X,fy.) be a semi-uniform space. Let A, 12

b) A £k(A )

c) if AsB; then k(A)^k(B)

d) k(AUB)=k(A)l/k(B)

e) k(A)gc(A)

Proof: a) Clear.

b) Let a eA® Since A £. U for every U e^JL , a e UCaJA A

for every TK'U . Thus aek(A).

c) Suppose A<£B and xek(A). For every

UelL ,jif^Utx]AAc{J[x]AB. Thusxek(B).

d) By (c), k(A)Uk(B)^k(AUB). Let xe k(AUB). If

x f k(A)Uk(B), 3 U e such that U(_x3A A= and

3 V ell such that V[x]/)B = Then

(U/1V)[xJA(AUB)=^ contradicting xek(AUB). Thus

x £ k(A)t>k(B).

e) Let xek(A) and let xe 0 6 Then 3 U € V. such

that U£xO£0. Since U fxJ/lA ^, then O/lA^k^. Thus

x e c(A ).

I. 1.29 Theorem. Let (X,^.) be a semi-uniform space and let

ASX. Then k(A) = A fuj>] )u£^^.

Proof: Let xek(A) and let V eil . There exists a symmetric

V e°U. such that VSU. Since Y[x]/"| A,£^, 3 aaA such that (x,a)e V.

Then (a,x)eV£U so that xeU[A]. Let xe /3 [u[A]\u , l e t U e U and l e t VellU. be symmetric with V£U. Since x

~~I. 1.30 Theorem. Let (X,K.) be a semi-uniform space and let~~

~~F£X. Then F is closed in (X,^(^)) if and only if k(F)-Fe Proofs Suppose F is closed. Then, using 1.1.28,~~

~~F£k(F)£c(F)nF.~~

~~Now let k(F)=rF and let x cCF, By 1.1.29, 3 a symmetric~~

~~U such that x^UfF]. Then U(XJ<^£F. Thus £F and F is~~

~~c lo se d .~~

~~The next few lemmas lead to a characterization of the semi-~~

~~uniformizable topological spaces.~~

~~I. 1.31 Lemma. Let (X/U) be a semi-uniform space. Then, for~~

~~every x eX, k{x} £ C\ { o jx £ 0 &~~

~~Proofs Let xtXj let xtO e-^ ) . Then there exists Ve'U such~~

~~t h a t V[x]Q0» Since k[x} = f\ [u[x] )u G-ilu } s k{x}G V(.'xJ^ 0.~~

~~I. 1.32 Definition. Let (X,1^) be a topological space. Let~~

~~-[A^xtX^be a family of subsets of X. ^A^xe X? is an open-~~

~~determining family provided that the following holds s 0* Q,*X. i f and~~

~~only if x£ 0* implies 3 0 £ and a finite subset F£X such that~~

~~xeO-FgO^.and x^Ay for every y^F.~~

~~While the last definition is unusual, the next lemma shows that~~

~~open-determining families arise naturally in semi-uniformizable~~

~~topological spaces.~~

~~I. 1.33 Lemma. Let (X,“U) be a semi-uniform space. For each~~

~~xeX, let Ax=-kfx}. Then ^A ^xeX ^is an open-determining family for~~

~~(X,W)).~~

~~Proof: Let 0 € 't (i(). For each xeO, let F=^. Then xeO-PsO~~

~~and yaF implies x^A • Let 0*^X and suppose xeO* implies 3 y •8* / and a finite subset F£X such that xc 0-F£0 and xf A for every ,~~

~~U = ( /ifU y/y £F^)/^Uo0 Since this intersection is finite, UfeK.~~

~~Then U£xj£ 0-F£0*. Thus 0 *6^ (U),~~

~~I. l»3h Lemma. Let (X/O be a topological space; let~~

~~Proof; By 1.1.23, it is only necessary to show that *T (^0.~~

~~L et 0* G ^(U) and let xeO*. Then 3 x^, o ..,^£ X and 0]_, ...,0 such that A Sx^i0j>CxD -0^• By 1.1.21;, A SXij0^[xlrrO-F where~~

~~0 G°t and F is a finite subset o ff1 x^, ...,x ^ . Moreover, by 1.1.21;,~~

~~Xj_£F if and only if xÔ^. Thus, since A^cO^ for each i, x^ê F implies xÂ ^. By the defining property of an open- determining family,~~

~~I. 1.3^ Theorem. Let (X,^) be a topological space. Then (X,^) is semi-uniformizable if and only if there exists an open-determining family { x £ X^ such that Ax^a£o)x£ O e 't'l for each xeX.~~

~~Proof; 1.1.31; shows the sufficiency of the condition. On the other hand, if V is a semi-uniformity such that ‘TCV )='*£', let~~

~~Ax=kfx£. The remainder of the theorem now follows from 1.1.33 and 1.1.31.~~

~~I. I.36 Corollary. Let (X/t) be a topological space and let it be the semi-uniformity as in 1.1.23. Then (X,°£) is semi- uniformizable if and only if CX.~CZ(ti).~~

~~P ro o f; Assume (X/<0 i s sem i-uniform izable and l e t °li* be a 15~~

~~semi-uniformity which generates 'T „ For each x, let Ax— k*{x]' 0~~

~~Then, applying 1.1.31* 1*1.33 and I.1.3U* — ‘Tfyi) • The converse i s~~

~~c le a r .~~

~~I. 1.37 Corollary. Let (X^T) be a semi-uniformizable~~

~~topological space. Let ^eX . If A*[o|xoe 0 is finite, then~~

~~A - f o ) x 0e 0~~

~~Proof; Let F*= A{o|xq£0 e and let 0^= C A By 1.1.31;,~~

~~3 an open-determining family { A IxeX^with A Q /^loJx e 0£ ^ fo r "■ X each x. Since F* is finite, F* — xQ,x^,...,xn . Let ye 0*. Then~~

~~ye. X-{xQ,X p.. .,Xh] = 0* and, since A^S. F* for each i, y^ Ax^ for each i. Thus 0* .~~

~~I. 1.38 Corollary. Let (X/^) be a topological space with X~~

~~finite. Then (X/*) is semi-uniformizable if and only if (X/V) is~~

~~s ta b l e .~~

~~Proof; This immediate from 1.1.22, 1.1.37 and I.1.20e.~~

~~It is interesting to note that this last fact becomes false if~~

~~"compact" replaces "finite". An example w ill appear later.~~

~~The next example shows that a family { A^ x e X^ as in 1.1.35 need not arise via the weak closure of some semi-uniformity.~~

~~I. 1.39 Example. Let X be infinite and let A be a proper infinite subset of X. Let *'T-=£o|0 or ASO^. For each xeX, let~~

~~Ax= A. Clearly, A^c H {o |x £ 0 £ T 1 for every x. is an~~

~~open-determining family for ^ as follows; Let xeO eT . Taking~~

F = ^, xeO-FsO and ye F implies xÂ . Now let X and suppose that xe 0* implies 3 a finite set F and 0 such that xeO -F irO * and x Â y f o r each y e F . Assume 0 * ^ and let x£ 0#, There exist 0 C°X and a finite set F such that XG.O-FSO* and xÂy for each yeF. Since A£0 and F is finite, there exists zeA/10# and, for z, there exist O’ and F* finite such that zcO'-F'sO* and z

~~Now l e t %i be any semi-uniformity with . For any x^A, since fx^ is closed, k( jx |)—fx^. For such x, k(^xj )f\ A^=- $<,~~

~~I . 2. Semi-Uniform Spaces With Property K.~~

~~The example which opens this section shows that three basic uniform space theorems fail in the more general context of semi uniform spaces.~~

~~I. 2.1 Example. Let X={1,2,3} «~~

~~Let B= AL/{(1,2), (2,1), (2,3), (3,2)|o By 1.1.5, 3 a semi uniformity 2^1 with base B .~~

~~a) k(k jl3)^k{l}. (Thus k is not the closure operator~~

~~in (X,Y(«0)0 By 1.1.29, k[l}= B[l] ={l,2] and k£l,2l=Bfl,2l]=fl,2,3l.~~

~~b) Let T={l,3l and let V = Y*YAU . (Y,^CV)) is not~~

~~a subspace of (X ,^(^)): Since B[l]—{1,2},~~

~~B[2]= fl,2,3} and B [3]-{2, 3}, it is clear that~~

~~Moreover, Bf\Y*Y — Ay so that -fAyl is~~

~~a base for V . Thus, (Y,^CV)) is discrete.~~

~~c) B[l] is not a neighborhood of 1: As above,~~

~~^(20={^,x} and B[l]={l,2}.~~

This last example, together with 1.1.28, shows that, if k is a weak closure associated with some semi-uniformity, k satisfies 17 all the properties used by Kuratowski to characterize topological closures except (perhaps) k(k(A))= k(A)» This leads naturally to the next definition,

~~I. 2.2 Definition. Let (X,V.) be a semi-uniform space. (X,%) has property K if and only if k(k(A))=k(A) for every ASX.~~

~~I. 2.3 Theorem. Let (X}U) be a semi-uniform space. Then (X,U) has property K if and only if for every ASX, k(A) = c(A).~~

~~Proof; Suppose (X,%) has property K and let AsX. By I.1.28,~~

~~ASk(A)£c(A). Since k(k(A))= k(A), k (A) is closed by 1.1.30.~~

~~Thus c(A)£k(A). Suppose now that k(A) = c(A) for every ASX. Then k(k(A))=c(c(A))=c(A) = k(A) for every ASX.~~

~~I . 2,h Theorem. Let (X,U.) be a semi-uniform space. Then (X,U) has property K if and only if UCxJ is a neighborhood of x for each x£X and each U e% .~~

~~Proof; Let (X,&) have property K, let U <£ *U and let xeX.~~

~~Let A = CU[x7. Then, since x^k(A)=c(A). Thus x e. (5cA^U(V]. On the other hand, suppose U[x] is a neighborhood of x for every x £X and every U e il, Let ASX. By 1.1.28, k(A)*== c(A).~~

~~Let xccA. For every Ue'R, U£xJ a neighborhood of x so that~~

~~Thus xek(A).~~

~~I. 2.^ Corollary. Let (X/U.) be a semi-uniform space. Then~~

~~(X,U) has property K if and only if, for every ASX,~~

~~Int A= j'xj U[x]SA for some UcttJ •~~

Proof; Suppose Int A=|x)u£x3SA for some U e ■a? holds for every ASX. Then, given U and xe X, x e l n t U£xJ. Thus (X,^) has property K. Now suppose (XstQ has property K and let ASX. For any 18 xGlnt A, 3 U such that U[x]^Int ASA. Thus

~~I n t A£=|x ]u [xlI£LA for some U If U[x]~~

~~I. 2,6 Corollary. Let (X/fc*) be a stable topological space.~~

~~Then (X, ^(^)) has property K.~~

~~Proof; Since *£( %.&))■=*£ and U.M is the set of neighborhoods of A , clearly Ufx] is a neighborhood of x for every xeX and Ue % ? £ ).~~

~~The last corollary im plicitly contains the fact that a semi uniformity with property K need not be a uniformity.~~

~~I. 2.7 Example. Let %L be the semi-uniformity for R constructed in I.1.16. Recall that V. has a base of the form~~

~~[v £ A B |t>0 and that Va [ x] and BtX^are -neighborhoods of x for each x and each £ ?0 . Thus, applying 1.2.It and the fact that~~

~~(V£ A B)[x}= [ x j AB [x ], (R,%) is a semi-uniform space with property K. However, as was shown in 1.1.16, and~~

~~^i')) ^.°ii o In particular a semi-uniformity with property K is not necessarily a subset of the neighborhoods of A .~~

~~I. 2.8 Example. Let Xsjajb^ and let >[a}, X^. Then t ( =|xxXjL Thus (X/T) is not stable and (X, WX)) has property K.~~

~~I. 2.9 Example. Let Xs-^ajb,^ and let~~

~~^=^,-[a^,^a,b}, fa,c]r, X^. has base /B^ where~~

~~B- fa,b}xâ,b} L>-[a,c}x â,c£ . B[c3 = â,c^ and B [a]-â,b,c^ so that k(k/c5)^kjc5. Thus (XfllCt)) does not have property K.~~

~~I. 2.10 Theorem. Let (X/£) be a topological space. Then~~

~~(X/fc) is stable if and only if there exists a semi-uniform space ^ 19 w ith p r o p e r ty K such that rT(‘k)~ ^ .~~

~~Proof: The necessity of the condition follows from the~~

~~definition of stable and 1.2.6. Now let %L be. a sem i-u n ifo rm ity w ith property K such that . By 1.1.31# for each xeX,~~

~~c^'x^kfxI^/O l'oJxeO e'T j. Thus (X/fc“) is stable by I.1.20e.~~

~~The next few results concern subspaces; in particular, the subspace difficulties pointed out in 1.2.1 disappear in the context of semi-uniform spaces with property K.~~

~~I. 2.11 Lemma. Let (Y,V)£~ (X,?0 be semi-uniform spaces. Then~~

~~(A) Y/1 (A) for every AS Y.~~

~~P ro o f: For A s Y, y e k y (A)^>y£ Y and (Y*Yft U)[$Afor ev ery~~

~~U C.V.~~

~~^ y £ Y and U£y]A A£ for every U 6 'U~~

~~^yeY /lku(A)~~

~~I. 2.12 Theorem. Let (Y,^)^ (X,U) be semi-uniform spaces. Let~~

~~(X,%) have property X. Then has property K.~~

~~Proofs Let A«E Y. Then, using 1.1.28 and 1.2.11,~~

~~kv (A )^k^(kv (A))= kv (Y/}k^ (A))~~

~~= YAk^(YAky(A))~~

~~£ Yf>k^(Y)Ak^ (ku (A))~~

~~~ YAku (A)=k^(A)~~

~~Thus ( Y,V) has p ro p e rty K.~~

~~I. 2.13 Theorem. Let (X/U) be a semi-uniform space and let~~

~~Y £X . Then YrfC(U) £ ^ (Y * Y A U ) .~~

~~Proof: Let ycO’e Y C\T(U)9 Then 0*=YA0 for some O ^fcO .~~

~~Thus there exists Ue ‘Us U£yl~~

~~so o'e^CYxY^U). I. 2.llj Theorem,, let (X,tO be a semi-uniform space and let~~

~~Y£X; let V= YxYr\U o Then (YS^CV))£(X,^C&)) if and only if~~

~~A—YAk^ (A) implies A=YA c^(A ).~~

~~Proof: Suppose ‘T (V) = X AT(fy) and let A=YAk^(A). Then~~

~~A£Y and k^ (A)= Y/l lc^ (A)^ A. Hence, by I d .30, A is closed~~

~~relative to <^'(ty)s= Y so that there exists F closed in~~

~~(U) such that A= YAF. Then c^(A )e.F and so A«=LYA c^(A)£ YAF=A®~~

~~Now suppose A-YA k^(A) implies At;YAc.^(A) and le t A be closed in~~

~~^ (y ). Then A= k^y (A) =rYnkcy (A) so that A=YftC^(A)« Thus A is~~

~~closed in Y (2i) and so ^ (V)<^ Y/A *^(20 • Thus, u sin g 1 .2 .1 3 ,~~

~~(Y,r(V))e(X,'T(^)).~~

~~I. 2.15 Corollary. Let (X,“iO be a semi-uniform space and let~~

~~YSSX be closed in (X,^(&)). Let V= YdTVU . Then~~

~~(Y ,w ))s(x,ra»- Proof: Let A=YAk(A). Then A^Y and so k(A)sk(Y)=Y. Thus~~

~~k(A) = A and A = YAc(A). By 1.2.11; (Y,^*CV))^(X,-T(^)).~~

~~I. 2.16 Theorem. Let (X,%) be a semi-uniform space. Then (X,%)~~

~~has property K if and only if (Y,^(YxY/lfy.))£- (X,*^^)) for every~~

~~YSX.~~

~~P ro o f: Suppose (X,V) has property K, le t Y<= X and let~~

~~A-=YAk^(A). Then by 1.2.3 A^YAc^A) so that~~

~~(Y, ^(YxY/1^ )) Sr(X,AT(£0). Now suppose that~~

~~• (Y,^'(YxYn^))c(x,^(‘^)) for every YSX. Let A<=X and let~~

~~Y= X-(k^(A)-A). Clearly, A = YAk^(A) so that, by 1.2.11;,~~

~~A=YAc^(A). Let xec^(A), If x^k^(A), then x£Y by the 21~~

~~definition of Y. But then x g YPI c^ (A)= A£k^(A). Thus~~

~~k «^(A)=: c ^ ( A ) and so (X,fy) has p ro p e rty K.~~

~~The next theorem contains a property which can be considered as~~

~~a weakening of the triangle inequality for semi-uniform spaces with~~

~~property K.~~

~~I. 2.17 Theorem, Let (X,fy) be a semi-uniform space. Then~~

~~(1M) has property K if and only if for .every U and xe X3 VelA.~~

~~such that yt£V(x] implies 3 Uy£% with Uy£y]c U£x]o~~

~~Proofi The condition is shown to be sufficient as follows: let~~

~~ASX and let xtk(k(A))« Suppose 3 U £*U such that U[x]r\A=^. Let~~

~~MeK be such that yeV£x} implies 3 Uy£ ‘U. with Uy[y3£U/XJ. Thus~~

~~y e V£.x] im p lies Uy[yj/) A= ^ so th a t Vfxlfl k (A )= ^ . But th is~~

~~contradicts x£k(k(A)). Thus k(k(A))=k(A) and so (X/U) has property~~

~~K. Now suppose (X,^) has property K, Let Ue'U and xeX, Since~~

~~UCx] is a neighborhood of x, 3 0 such that xeO£U£x]. Then~~

~~3 V such that V[x]€0, For each yeVfusf y6 0 so that 3 Uy£% w ith Uy[y3& 0 £UCx3 .~~

~~The next example points out a significant difference between~~

~~uniform spaces and semi-uniform spaces with property K: there may be~~

~~several semi-uniformities with property K which generate a given~~

~~compact, stable topological space. As a result, f:(X/£'(20)'-*(X/fctV)) with (X,*T(«0) compact need not imply f:(X,'20-»(Y,'V) for semi-uniform~~

~~/ spaces with property K.~~

~~I. 2.18 Example. Let (R,U) be the semi-uniform space~~

~~constructed in 1.1.16. By 1.1.16, (ii) is the usual topology for R~~

~~and by 1.2.7 (Rj^O has property K. Let X= [0 ,lj and let °V — X*Xf)lL . 22~~

By 1.2*12 (X,Y) has property K and by 1.2.16 ^ (V) is the usual topology for [0,l]. Recall that B (consisting of the x-axis, the y-axis and the points between the lines y=ijx and y=2x) is an entourage in °li • Then clearly XxXAB is not a neighborhood of (0,0) and so V is not the usual uniformity for X.

~~The next few results point out some situations in which uniformly continuous maps preserve property K.~~

~~I. 2.19 Example. Let (X,y) be a semi-uniform space without property K. Let V— {'xxxj and let Y/ be all supersets of A . Let~~

~~IsX-S-X be the identity map. Then I:(X/U)—»(X/V) since and~~

I:(X,V)-»(X/U) since . Both (X/V) and (X,V) have property K while (X,*U) does not*

~~It is worth summarizing the last example as follows? Let f: (X,fy)-? (Y/Y) be 1-1 and onto. Then (Xyfr) may have K without (Y£V) having K and (Y/V) may have X without (X/U) having K.~~

~~I. 2.20 Theorem. Let (X/K) and (Y/V) be semi-uniform spaces and l e t f : (X/U)~*( Y/y) be onto. Suppose that f is open relative to~~

~~(U) and‘T(V) and that (X,U) has property K. Then (Y,i/) has p ro p e rty K.~~

~~Proof: Let V ey and yeY. Then 3 xeX such that f(x) = y.~~

~~Let U=(f*f )“*!>]. Since (X/R) has K, a 0 such that x £ OSUCxl. Then y=f(x)e f[0] £ f[utxi] • Since f is open it is sufficient to show that ffutx]]^ Vfy]. Let y‘e f^UtxQj . Then~~

~~3 x'fcUJx] such that yft=f(x'). Then (x,x')eU and so~~

~~(f(x),f(x‘))= (y,y')£V. Thus y*£ Vfy],~~

~~I. 2.21 Corollary. Let (X/U.) and (Y,V) be semi-uniform spaces 23 and le t fs (X^)-^(Y/V) be a unimorphism. Then (X,*W) has property K if and only if (Y/y) has K.~~

~~Proof; Using 1.1.15, both f and f '* are uniformly continuous, open and onto. Thus the result follows immediately from 1.2.20.~~

~~I. 2.22 Lemma. Let (X/U.) and (Y//) be semi-uniform spaces and let f :(X,fy.)-*(Y/V). Then f[k ^ (A)] £ k^ (f[A]) for every ASX.~~

~~Proof; Let y£f jk^(A)J and let Ve V • There exists x£k<^(A) such that y=f(x). Let U= (f^f)-1 [vj. Then U[x3flAjfc^ since x£k^(A ). Let aaUCxjflA. Then (x,a)aU so that~~

~~(f(x),f(a))= (y,f(a))<£ V. Thus f(a) e VCyJ/1 ffAl and so y ck v (ilA]).~~

~~I. 2.2 3 Theorem. Let (X/U) and (Y,V) be semi-uniform spaces and let fs(X/H)-*(Y/V) be onto. Suppose that f is closed relative to~~

~~(X,T(2t)) and (Y,0f CV)) and that (X/U.) has property K. Then (Y,V) has property K.~~

~~Proof; Let AS X. Then k^(A) is closed and so f[k^(A)J is closed. Thus, since f[A ]£f[k^(A )], k~~

~~1.2.21 f [k^ (A)|= key (f[A] ). Now let BsY and le t A=f~f[B], Then f [aQ — B and f[k ^ (A )|- k~~

~~.ki/(k1/(B ))=kv (f[k%(A)]') = f[k%k~~

~~(Y,V) has property K.~~

~~I. 2.20 and I. 2.23 naturally suggest the following question:~~

~~I f f : (X,ii)-^(Yf/ ) , f : ( X , (1t))~^(Ys~~

~~(Xj'U) has p ro p e rty K, does (Y/Y) have K? The next example answers this question negatively.~~

I. 2.2U Example. Let (X/U) be £ o ,lJ with the usual uniformity; let Y=£a,b,c^ and let be the semi-uniformity for Y with base B= ■j'a,b3*-[a,b^\J ■£b,cj’*{b,c^. As in 1 .2 .9 ,, (Y,^) does not have K and (V )=|V ,xj. Now define fiX-^Y by r a i f xe (OjtJ

~~f(x)=y b if x~~

~~V c if x € [%,!)~~

~~Clearly, f_,-fa}, f -,^b} and f“*{c} are not open in £o,l]. Also f"Ya,b5=[0,^v)t;{l}, f"'fa,c?= (0,$]U f|,l) and fb,c^=-{o} U (^fl). Thus (Y) is the quotient topology from f and~~

~~^(U ). To show f s (X/U)-»(Y/V) it is sufficient to show that~~

~~(fxfy^tB]] is a neighborhood of A . £ 0,^) x£o,ir)^= (f* f)~ ^Isin ce f(x) is either a or b for xe[o,f;) and fa,b$x|a,b^£=B. Similarly,~~

~~,l]<=(f*frJ£B]' The next example shows that, given f s(X,U)-*(Y/V0* hypotheses stronger that those in 1.2.20 and 1.2.23 w ill not guarantee that~~

~~(X/U) has K when (Y/Y) has K.~~

I . 2.25 Example» Let X = {a,b,c^ and let °IL be the semi- uniformity with-{a,b3x|a,bj U fb,c}x^b,c3 as base. Let =^XxX^. As above, *'£,(%) = ;[/,X ^ = ‘£'(‘Y). Let I:X"*X be the id e n tity map. Then

~~I:(X,‘U)-J>(X/V'),I:(X,‘yt'(‘W))->(X,^(Y)) is a homeomorphism, (X,V) has property K and (X,*iO does not.~~

~~The results which conclude this section give a condition which, together with uniform continuity, makes it possible to conclude that the domain space has property K if the image has property K.~~

~~I. 2.26 Theorem. Let f:X-^Y and let °V be a semi-uniformity for Y. Let |(fxf r/O Js X«X for some V c V ] » Then^C i s~~

~~i the smallest semi-uniformity for X such that f ;(X,Y)—KY/Y). The proof of the last fact is standard. From this point on* given (Y//) and fsX—»Y, f_,~~

~~I. 2.27 Lemma. Let (Y/V) be a semi-uniform space with property~~

~~K and let f:X~»Y. Then U[xJ is an f ^(^-neighborhood of x for every Ufcf_,V and xeX.~~

~~Proof; Let xe X and Uef~,cV . Then 3 V 6 ^ such that~~

(fxf)”1 [v3—U. Since (Y,V) has property K, 3 O^'TfV) such that f(x)e 0£-v[f(x)| . Since xe f~,fo j£ f“/|V[f(x)jj , it is sufficient to show f ~l jy [f(x)]J<^ U£x]. Let x*£ f “‘^vjf(x)]j . Then (.f(x),f(x')) e V and so (x,x')£U . Thus x'e U£x]«

~~I. 2.28 Theorem. Let (Y/V) be a semi-uniform space with property K and let f;X-*Y. Then * C ( f )= fand (X,f “'V ) has p ro p e rty K.~~

~~Proof; Since f;(X, f*W )-»(Y,tV), f;(X/?(r'V ))~KY,W )).^~~

~~Thus f "‘V ). Let x f;0 e ^ f 'Y ). Then 3 Ue f " V such thatU [xj£0. Thus, using 1.2.27, Oef”1^ (V). Since f ~,TeV) = ^ (f~ V ) and 1.2.27 holds, it is immediate that (X,f“*V) has property K.~~

~~I. 2.29 Example. Let X=-ja,b3 and let Y=-[a,b,c^. Let 6V be the semi-uniformity for Y with base a,c3x{a,cj t/-[b,c}}i|b,c^. As before ^(V ) is indiscrete. Let f;X-^Y by f(a)=a and f(b)=b. Then~~

~~£~,eV has base (fxf)''[B] — -fapc-fa^Ujb}xJb}, Then ^ ( f "'Y ) i s discrete. Clearly, £~%*T(y) is indiscrete and ^(f^fy )^4f “^CV).~~

~~I. 2.30 Theorem. Let X be a set and let (Y,aA be a semi uniform space. Let f;X-»Y be onto. Then c^'(f*’V V). Proof; Since f: (X, ^(f "'7 ))-*(Y, W )), f ~'^C V ^^C f'V ).~~

~~L et 0 <=^(f~leY ) and let xeO and let y = .f(x). There is a Ue f ^ \ / such that U[xj£ 0 and 3 V 6 such that (f*f)~' [vjsu. Let y'sVfyJ, There is an x1 e X such that f(x,)=.yl. Then~~

~~(f(x),f(x'))£. V and so (x,xf)^U. Then x'^0 and so y'e f[dj. Thus~~

~~V['y]£f[0] and f [0] £ ^(V). Now it is sufficient to show that~~

~~O r f 1 [flol] . It is always true that 0 £ f ' iJf [0^. Let x ef"1 jjffojj .~~

~~Then 3 x'£ 0 with f(x')=:f(x). There is U ef^V such that~~

~~U [x'j£0 and 3 7 eY with (f*f )“' (>]c u. Then (f (x'),f (x))e V and so~~

~~(x'jxje U. Thus x eUCx’j £ o.~~

~~I. 2*31 Theorem. Let X be a set and let (Y/V) be a semi uniform spacej let f:X-»Y be onto and let ‘U. — f ~*V . Then (X/U) has property K if and only if (Y/V) has property K.~~

~~Proof: If (Y/V) has property K, by 1.2.27 (Xyil) has property K.~~

~~Now suppose (X/IQ has property K. Let V £“V , let yeY and let~~

~~U= (fxf) *[v], Then U and 3 xeX such that f(x)=y. Since (X/H) has p ro p e rty K, 3 0 £ such that xe OSUfx] . By 1.2.30~~

~~0 = f - '[ 0 'J f o r some 0 1 £ °Z(V )• Hence y = f(x)e f[o ]r o 's f juix}]s=i VCy] and so (Y fV) has property K. CHAPTER I I~~

~~THE HYPERSPACE OF A SEME-UNIFORM SPACE~~

~~II. 1. Definition And General Facts.~~

The hyperspace of a semi-uniform space is constructed in exactly the same way as the hyperspace of a uniform space. In the first few paragraphs the proofs omitted are exactly the same as in Caufield [ 3 ]•

~~II. 1.1 Definition. Let X be a set and let S£X*X.~~

~~H(S)-f(A,B)|A^X,B^XaA‘SSrB3 and B£S/A]j.~~

~~II. 1.2 Lemma. Let X be a set and let S,T£.X*X.~~

~~Then a) if then Ag G. H(S)~~

~~b) H(S)-'= H(S)~~

~~c) if S£T, then H(S)^H(T)~~

~~d) if H(S)SH(T) and S=S~', then SST~~

~~e) H(SAT)£H(S)nH(T)~~

~~f) H(S)*H(S)£H(S°S)~~

~~II. 1.3 Lemma. Let X be a set and let X>tX. Let~~

~~(i~fH(B)|B£ (8].~~

~~Then a) if (B is a base for a semi-uniformity, I) is a base for a~~

~~semi-uniformity A b) i f (p is a base for a uniformity, is a base for a~~

~~uniformity.~~

~~II. l.li Theorem. Let X be a set and let *U, and be semi uniformities for X with bases and respectively. Let be~~

~~27 28 the semi-uniformity for X with base ^ H(B^)|b^£ and let be the semi-uniformity for X with base £h(B 2)|B 2 6 Then ‘2^,= CU% i f <\ A and only if *Ut ~ 21 x .~~

~~II. l.£ Definition. Let (X/k) be a sem i-uniform sp ace. The~~

~~A A hyper space of (X/U) i s (X,fy) where il is the semi-uniformity with base~~

~~{ h (U )|u £ li].~~

~~Note that by II.1.1* any base for a given semi-uniformity w ill yield a base for the hyperspace semi-uniformity.~~

~~II. 1.6 Lemma. Let Y£X be sets and let SS=X*X. Then~~

~~H(Y»YnS) = y *y a h (s ).~~

~~II. 1.7 Theorem. Let (X/U) and (Y/K) be semi-uniform spaces.~~

~~Then (Y,V)c(X,$0 if and only if (Y/VO^ (X,4).~~

~~II. 1.8 Theorem. Let (X/U) be a semi-uniform space and let i:X"*X by i(x) = |x]. Then i:(X/2l)-»(i|jfl ,i[#x ) is a unimorphism.~~

~~II. 1.9 Lemma. Let (X/U) and (Y,V) be semi-uniform spaces and let (X/U.) and (Y,V) be unimorphic. Then (X/U) is a uniform space if and only if (Y,V) is a uniform space.~~

~~Proof: Clearly, it is sufficient to show that, if (X/U.) i s a uniform space, (Y/V) is a uniform space. Thus suppose (X/U) i s a uniform space, let f:X-»Y be a unimorphism and let V eV . Then~~

~~3 U e°U such that U°U^(f*f)"' Lv]. LetW = f*f[u]. Then We*V and it is sufficient to show that W°W£V. Let (y,y' )t VM/tf. Then~~

~~3 y " e l such that (y,y")eW and (yM,y')£Wo Let x,x’ and xm£ X with f(x)= y, f(x.,)=yl and f(x,,)= yM. Then (x,xu )£B and~~

~~(x*',x')eU so that (x,x*) £ (fxf)-i£v]. Thus (yjy'je'V# 29~~

~~II. 1.10 Theorem. Let (X,&) be a •semi-uniform space. Then~~

~~A A (X,%) is a uniform space if and only if (X,^) is a uniform space. »\ a Proof; If (X,%) is a uniform space, then by II«1.3b (X,^) is~~

~~A A a uniform space. If (X,%) is a uniform space, then, letting i:X~*X by i(x )—-[x^, (i[x],i[x]* i[x] A %L) is a uniform space. By~~

~~II. 1.8 and II. 1.9 (X,2{) is a uniform space.~~

~~II. 1.11 Definition. Let X and Y be sets and let f;X-*Y. The map f;X-*Y is given by f(A)— f|3Q for every k& X.~~

~~II. 1.12 Lemma. Let X, Y and Z be sets, let f:X—^Y and let g;Y-*Z. Then gof = g°f.~~

~~IIo 1.13 Lemma. Let X and Y be sets and let f:X->Y.~~

~~Then a) f:X->Y is onto if and only if f;X-*Y is onto~~

~~b) fsX-^Y is 1-1 if and only if f;X-*Y is 1-1 v ^-1 c) If f is 1-1 and onto, then f = f .~~

~~Proof; Suppose f is 1-1. If x^x', then so that fCfx^yAfCfx'?). Thus ^f(x)^ ¥■ ^*(x' Jj', that is, f(x)^f(x'). Now~~

~~A /\ suppose f is 1-1 and f(A)=f(B). Let aeA. Then 3 b cB such that f(a)?=f(b). Hence a=b and A^B. Similarly, B£A. Thus f is 1-1.~~

~~Suppose that f is onto and let y£Y. Then 3 A e $ such that f(A)~{y]. Then A^t^ since f(^)= /. For any aeA f(a)t=y. Thus f is onto. Now suppose that f is onto and let B el1. Let A=f“*fB].~~

~~Then f(A) = ff~’[ Bj= B since f is onto. Thus f is onto.~~

~~Now suppose that f is 1-1 and onto. Let I;X—*X be the identity map. Then I:X—»X is the identity map. Also f"'o f - I~ f f ^-f"^ f and so f ” * = f"^~~

~~II. 1.1U Theorem. Let (X,^l) and (Y,Y) be semi-uniform spaces 30~~

~~and let f;X-*Y. Then f : (X#)-*(Y,(Y5V).~~

~~II c lolS Corollary. Let (X/ii) and (Y,V) be semi-uniform spaces~~

~~and le t f :X-»Y. Then fs(X,(Y,'V) is a unimorphism if and only if~~

~~f :(XJti)->(Y,‘V) is a unimorphism.~~

~~Proof; This is immediate from II.1.13 and II.l.ll*.~~

~~II. 1.16 Corollary. Let X be a set; let V and V be sem i-uni~~

~~formities for X. Then if and only if »~~

~~Proof; Let I;X-*X be the identity map. Then IsX—»X is the~~

~~identity map for X. W I; (X/V)—’’(X/M.)~~

~~I;(x/v)-*(X,&) o U ^ V .~~

~~II o 1.17 Theorem. Let (X,*U) and (Y,°K) be semi-uniform spaces;~~

~~let f;X-»Y. Then U- f"'V if and only if "U=f “'V .~~

~~Proof; Suppose that V. - f _,V • Then by II. 1.lit ft(XjU)—^(Y/V).~~

~~Let V be a sem i-u n ifo rm ity f o r X such th a t f : ( X/W)-^ ( Y/V) „ Then~~

~~f:(X/k/)-?(Y,V) and so 2(=f''^V . By II.1.16, Thus~~

~~Vi - f ' 1 V . Now suppose that V- f _/~~

~~f;(X;U)-»(Y/V). Let . There exists BeV. with H(U)5U and~~

~~3 V c V with (fxf )~‘[v3<£ U. It is sufficient to show that~~

~~(f* f)H /h (V )J q H(U)<=U. L et (A ,B )e (f* fT '[H (V )] . L e t a s A . Since~~

~~(f(A),f(B))£ H(V), there exists baB such that (f(b),f(a)) <2. V. Then~~

~~(b,a)e (fxfr^vJsU . Thus a cU[B] and so A^.U[B]. Similarly,~~

~~BSUfAl. Thus (A,B)e H(U).~~

~~n . 1.18 Theorem. Let (XjK.) be a serai-uniform space, let Y~~

~~be a set and let f:X—5>Y be onto. Let f(iO -^v |(fx f) [u}<~ Ve Y*Y for~~

~~some U £ ilj • • Then £(%d is the largest semi-uniformity for Y which makes f uniformly continuous.~~

~~Proofs Since f is onto, for each U c= U Aj£(f*f)QJ]. Also~~

~~[(f*f )[U]]H - f«f (u"’J and (f^f)[u1n U2]«^ (f*f fl (fxf) [U2] .~~

~~Thus f(^) is a semi-uniformity. By 1.1.10 f : (X/U)-*(Y,f (*0) and f(?0 is the largest semi-uniformity making f uniformly continuous.~~

~~From this point on, given (X/U), Y and fsX-?Y onto, f(u) w ill denote the semi-uniformity constructed in II.1.18.~~

~~The semi-uniformity f('U) as in II .1.18 is the quotient semi-uniformity for Y induced by f and% . Note that, if is a uniformity, f(U) is not necessarily the usual quotient uniformity for~~

~~Y. (In fact, it is generally not a uniformity.) For more on this subject see Himmelberg £8].~~

~~II. 1.1? Definition. Let (X/U.) and (Y,V) be semi-uniform spaces and let f:X—^Y. The map f is a semi-uniform identification (relative to % and y ) if and only if f is onto and V — f(%).~~

~~II. 1.20 Lemma. Let X and Y be sets and let fsX-^Yj let~~

~~SfeX*X w ith S - S - ' . Then H(f~~

~~Proof; Let (A,B)e H(fxf[S])» For each aeA, 3 b eB such that~~

~~(b,a) a f*f [S] and so 3 (x^(a),x2(a)) e S such that~~

~~( f x f )(x1 (a),Xg(a)) = (b,a). Similarly, for each b<£B 3 (x ^ (b ),x ^ (b ))e S such that f(x^(b))eA and f(x^(b)) = b. Let~~

~~C ^ x ^ a ) Jae A^U^x^bjjb £B^ and let D= {x-j_(a)| a£ A}t/\ x^(b)| b £ B^.~~

~~Since f(xg(a))=-a and f(x^(b))cA, it is clear that f(C) = A.~~

~~Similarly, f(D) = B. Now it is sufficient to show that (C,D)eH(S).~~

~~L et c £ C. If c =Xg(a) for some aeA, then (x-j_(a),X 2(a))e S and so c£S(XU If c c x^(b) for some beB, then (x^(b),x^(b))e S~‘= S and so ceS[D]o Similarly, D£S[C]. Thus (C,D) H(S).~~

~~II. 1.21 Theorem. Let (X,ty) and (Y/VO be semi-uniform spaces and let f:X->Y. Then f is a semi-uniform identification if and only~~

~~A. if f is a semi-uniform identification.~~

~~A. A. P ro o f x Suppose that f is a semi-uniform identification. Then f is onto so that f is onto and V = f(&)• By II. 1.lit fs(X/2i)-^ (Y/V).~~

~~Let "V be a semi-uniformity for Y such that f s (X,^()-?(Y,V). Then f:(X,&)-»■( Y/ft) and so V s f ^ Y . By II.1.16 V . Thus~~

~~V = f(% ). A Now suppose that f is a semi-uniform identification. Then f is~~

~~onto and f : (X ,tt)—=>(Y,V) so th a t V £ f(il). L et V£f(U), Then~~

~~3 U ££U such that (f*f)tu]QV and 3 a symmetric U C'U. such that~~

~~H(U)SD. Then (f»f)£ltf € V and, by II.1.20,~~

~~H(f*f[U])cfx^[H(U)]£M[0]£V so that V^V . Thus *V=f(&).~~

~~H . 2. HK Spaces.~~

~~The discussion of semi-uniform spaces with property K and the~~

~~construction of the hyperspace of a semi-uniform space leads naturally to the following definition.~~

~~II. 2.1 Definition. Let (X,20 be a semi-uniform space. (X,%) is an HK space if and only if (X,^.) has property K.~~

~~Since the hyperspace of a uniform space is a uniform space, every uniform space is an HK space. The following example shows that an HK space need not be a uniform space.~~

~~II. 2.2 Example. For each neP let En=£mjm^nj and let Vn be the following subset of P^PsV'n=^(k,j?)|k = /? or ktl%,n and | k-J?|^ 2J .~~

~~Clearly, A£iVn, V =V"' and £°r every n. Let "V be the 33 semi-uniformity for P with base ^ Vn|n 6 P^. The fo llo w in g shows th a t~~

~~(P,15)/) is not a uniform space: Suppose that ^or sorne n°~~

~~Pick J? 5 n+3. Then (n,5)~~

~~But then (n,n+l) £ which is a contradiction. The following shows that (PjV) is an HK spaces Let A£P and let neP. By 1.2.17, it is sufficient to show there exists m e P such that, for every~~

~~BeH(Vln)[A], there exists k£P with H(Vjc)[b3- H(VnJ[A].~~

~~Case I s A is finite or empty. Then 3 meP such that m?n and aM =fA5-~~

~~Clearly, for every BeH(Vm)[A], H(Vra)[B]£H(Vn)CAl.~~

~~Case IIs A is infinite. Let m be the smallest element of~~

~~■^kjkcA and k^>n^. Note the followings Since me A, clearly~~

~~{m}UEffl+2$Vra[A] • Since A is infinite, m+leV^CAl. Thus~~

~~T JA > Al/Em and so Vm[vm£Aj]= vjiju vJ^A U ^U B ^ VjJQ. Thus~~

~~7mK W j= VmW- ^B eH O ^fA ]. If B is finite, as in Case I, there exists Ne P such that H(Vjj)£b[1 = {b|. Then~~

~~H(VN) [ B ] £ H(Vn t2)fA]S H(Tn)[A].~~

~~Now suppose B is infinite. It is sufficient to show that~~

~~BS.V^gCcJ, aeC. In particular, aeVnCcl. If al? m+2, as above raeC.~~

Then (m,a)£Vn and so aeV ^C ], Thus C£H(Vn)(X}*

~~II. 2.3 Theorem. Let (X,%) be an HK space. Then (X,%) has p ro p e rty K. 3U Proof; By I.l«8 (X/U) is unimorphic to a subspace of (X,%).~~

~~Since a subspace of a space with property K has property K and unimorphism preserves property K, the space (X/U) has property K.~~

~~The example in n.2»!> shows that a semi-uniform space with property K need not be an HK space.~~

~~II. 2.U Lemma. Let (X/U) be a semi-uniform space with property A /A A\ A Kj let k be the weak closure associated with (X/w.). Let AeX. Then k({A})={B|k(A)=k(B)J.~~

~~Proof: Let Belc(fAf). Then B£H(U)[A] for every U so that~~

~~B£ /Oju[Aj\u 6£l(|=k(A) and A Sn{u[B ]|u€^|sk(B ). Thus k(B)^k(k(A))=k(A) and k(A)Sk(k(B))?k(B).~~

~~Now suppose that ic(A)— k(B) and let . Then~~

~~B Sk(A )=nfu[A ](ue^]£ VUl and A£k(B) = /l{u£B]/U£i£$£VfB]so t h a t A e H(V) [B ]. Thus B e1c( {A] ) .~~

~~II. 2.£ Example. Let X be an infinite set and let be the~~

~~co-finite topology for X. (X/fc') is T-j_ and so (X, has property~~

~~K. L et A be an i n f i n i t e su b set of X. By I I . 2 .U,~~

~~1c({a|)=s£b|c(A)= c(B) = xJ. Thus k($Aj) consists of all infinite~~

~~subsets of X. Now let BSX with B^^. For any U ail (^T) U[Bj i s a neighborhood of every b£ B so that U03] is infinite. Thus~~

~~H(U)[b] f) lc(fAj)^/f and so Bek(k ({$))• As a r e s u l t , 1c(k({A}))~~

~~A A consists of a ll non-empty subsets of X so that (X/U) does not have property K.~~

~~I I . 2 .6 Theorem. L et (Y/V)f=:(X,2/.) be sem i-uniform sp aces; l e t~~

~~(X/U) be an HK space. Then (Y/V) is an HK space.~~

~~Proof; By II. 1.7 (Y//)£:(X,U) and so, since (X/U) has property 3£~~

~~K, (Y/V) has property K. Thus (Y/V) is an HK space.~~

~~IIo 2.7 Theorem. Let X be a set and let (Y/V) be an HK spacej let f:X-»Y and let U- f"V . Then (X/U) is an HK space.~~

~~Proof: By II.1.17 LU= f-,V . Since (Y,V) has property K, by~~

~~1 .2 .2 8 (X/ti) has property K. Thus (X/U) is an HK space.~~

~~II. 2.8 Theorem. Let X be a set and let (Y/V) be a semi uniform spacej let ^X-^Y be onto and le t U ••= f “1V . Then (X/U) is an HK space if and only if (Y/V) is an HK space.~~

~~Proof: If (Y/V) is an HK space, by II.2.7 (X/U) is an HK space.~~

~~Now suppose that (X/U) is an HK space. By II. 1.17 .~~

~~A. A A v A A Since f is onto and (X/U) has property K, by 1.2.31 (Y/V) has property~~

~~K. Thus (Y,V) is an HK space.~~

~~H . 2.9 Corollary. Let (X/U) and (Y/V) be semi-uniform spaces and le t f:(X,U)-i> (Y/V) be a unimorphism. Then (X/U) is an HK space if and only if (Y/V) is an HK space.~~

~~Proof: This follows immediately from II.2.8 since f is onto and~~

~~U- f “V .~~

~~The next example shows that the semi-uniform quotient of an~~

~~HK space need not be an HK space.~~

~~II. 2.10 Example. Let X-^,b,c,d| and let Y=£a,b,c^. Let f:X*-^Y by f(a) = a, f(b) = b, f(c)=b and f(d)=c. Let~~

~~B={a,b|x{a,b| U{ctd.JxfctdJ and l e t £IA. be the semi-uniformity with base~~

~~{ B j. °IA- is a uniformity since B is an equivalence relation. Thus~~

~~(X,U) is an HK space. By II. 1.18 the quotient semi-uniformity has base (fxf)[B]si £a,bj;<£a,b^U Jb,c3*£b,c^. Thus ($,f(£d)) does not have property K and so (Y,f( )) is not an HK space. 36~~

~~II. 2.11 Definition,, Let (X,U) be a serai-uniform spacej let~~

~~A £ X and l e t U e U . A (U )= {b | a £U [B ]^.~~

~~The next theorem gives a characterization of HK spacesj the theorem leads directly to a characterization of the topological spaces which arise from HK spaces.~~

~~II. 2.12 Theorem. Let (X,U) be a semi-uniform space. Then~~

~~(X,20 is an HK space if and only if for every Ae X and U C'M.~~

~~1) 3 V £U such that VoV[A] ^U[A]~~

~~2) A(U) i s a *¥(&)-neighborhood o f A.~~

~~A. Proof: Suppose that (X/U) is an HI? space. Let AeX and let~~

~~. Since H(U)[A]£A(U) and (X,U) has property K, the set A(U) is a neighborhood of A. By I. 2.17, d( W£ V- such that for every~~

~~Be H(W)[Aj 3 UBc^U with H(Ub)CB_]ch(U)£a]. In particular,~~

~~W[A]£H(W)[A] so that 3 V 6 with V£W and H(V)[w[Al]^ H(U)fAj.~~

~~Then v[W[4) £ H(U)[ a ] and so VoV|A]c v[WfA]]s U(X].~~

~~The conditions are shown sufficient as follows: Let U £ and let A £ X. Let W e ^U. be such that ¥«W[A] U[aJ . Since A(U) is a~~

~~(&)-neighborhood of A, there is V £ % with V£ W such that~~

~~H(V)[A] C. I n t A(U). L et B £ H(V) [A] . Then B 6 I n t A(U) so th a t~~

~~3 Ub £~~

~~C £ A(U) so th a t A£U[C] . Moreover, C€U LbJ £ UB[v [A j}^ WoW[A] U[a] .~~

~~Thus for every B£H(V)[Aj[ ^ UB £ “U with H(Ub)[b3^=.H(U)£a]« By~~

~~1.2.17, (X,U) has property K.~~

~~II. 2.13 Lemma. Let ( X/U) be a semi-uniform space with property Kj suppose that for every A <£ X and U £ 3 V £ * li. w ith~~

~~V*v[a] £ U[A]. Let 0,0!£ *^($1) and suppose there exists U £*U.with with u[c(0)]co\ Then 3 o"e TQl) and WgU with~~

~~W [c(0)j£ 0' c(0' ’ )S W[c(0 '' )]0'.~~

~~Proofs By hypothesis there exist W, Vg and V-j_G.iL with~~

~~W^V2<=V1# V^V (c(0}]c u[c(o], V2oV2[c(0)]c vJcCojjand~~

~~WoWjc(O)] <^. V2[c(0)J . In p a r tic u la r , Wc-Wc-Wow£c(0)]<^ u[c(0)} . Since~~

(X,i() has property K, there exists 0M£ ^C^Jl) such that w[c(0)]£0‘'^Wow[c(0)] 0 By I elc29 c(0,,)£w£0ujso that c ( 0 " ) £ w [ c ( 0 , *)Jo W*w[o' '^¥^W oW oW [c(0)]cu[c(0)J. Thus w [ c ( 0 ) ] c 0 U £ c (0 " ) £ w [ c (0 " ) ] g 0 I .

~~II. 2.11* Lemma. Let (X,%) be a serai-uniform space with p ro p e rty K; suppose that for every Ae X and Ue'K 3 V &°LL w ith~~

~~7»VjjfJ U[a] . Then (X, ^ (&)) is completely regular.~~

~~Proofs Let xeO e^(it). It is sufficient to find a family of~~

~~open sets^ 0 ^ j 0<-^< 1^ such that < im p lies~~

~~xeOjx^S c(0^,)£0y4c c(0y8)s0. As usual, such a family is~~

~~constructed by induction: the construction of Oi,Oi and 0^, w ill 4 2 illustrate all the necessary techniques.~~

~~Since xe 0, there exists U efy. such that U[x]£=0. As in the~~

~~p receding lemma 3 V £% with VoVoV(x]£U[;x]. There is an~~

Ox.&cXfyQ with x£ 0i£V[x]. Then 2 2 x£0i S C(0i)sv[c(01)]^Vov[0i l^ VoVoV£x]^U[x]£ 0, that is, 2 2 2 2 xe0j.^c(0i)cvfc(0i)k 0. By the same process 3 and 2' 2' L 2 4 4 V-€.°IL such th a t x e O^ScCO^)£=¥-, ^0(0^)]^= 0^. By lemma I I . 2.13 1 4 4 4 4 applied to v[c(0i)]

~~Summarizing, 38 x e Oi£ c (0^)SVx[c(0^)]C c( ) c . W [c( o ^ ) ] c = O ^ G c(0^w [c(0^]s 0.~~

~~II. 2.If? Theorem. Let (X/¥) be a topological space. Then there exists an HK space (X^) with if and only if (Xft) i s completely regular.~~

~~Proofs Suppose 3 an HK space (X/U) w ith (&0 — « By II.2.3~~

~~(X/U) has property K and by II.2.12 for every AeX and~~

~~U e U 3 V £°U with VoV[A]Q U[A]. Thus by II.2.1k (X/V) is completely regular.~~

~~Now suppose (X/£) is completely regular. There exists a uniformity %( fo r X w ith Then (X/U) is an HK space with~~

~~II. 3» Topological Properties of the Hyperspace of a Semi-Uniform~~

~~Space.~~

~~The following definition w ill fix convenient notation:~~

~~II. 3-1 Definition. Let (X/T) be a topological space and let~~

~~B eX . B ^ I a I a s b ] and B“= £ a | a A~~

~~As is well known, given a topological space (X/V), the family~~

[o*! 0 is a base for Kuratowski's topology of upper semi continuity and the family ^0~|o is a subbase for the topology of lower semi-continuity. The next two results connect these topologies and the topology which arises from the hyperspace of a semi-uniform sp ace.

~~II. 3.2 Theorem. Let (X/U) be a semi-uniform space and let~~

~~B£X. Then B~£ * ? ($ ) if and only if Be*c(U).~~

~~Proof: ("^F ) Let Ae B“. ThenAflB^/rf so that 3 a c A AB.~~

~~Since asB, there is with UJa]£B. Let C£H (U )[ a }. Then 39~~

~~3 ctC such that (c,a)£ U. Then ceUfaj^B so that CAB^^. Thus~~

~~H(U)[ a ] c b - and B“e ^ ) .~~

~~(•=?) LetbeB. Then £b]e B“ and so 3 U=U- /g ^C such that~~

~~H(U)[{bljQB". For any aeD[b], {a}£ H(U)[{b£] and so {a]A B ^/,~~

~~Thus U[bj£B.~~

~~II. 3.3 Corollary. Let (X/%) be a semi-uniform space. Then~~

~~Proof; This is immediate from II.3.2.~~

~~II. 3.1; Corollary. Let (X,*U) be a semi-uniform space and let~~

~~F be closed in^(2i)» Then F'*' is closed in~~

~~Proof; Clearly F*=(fg( (f^F)”" and 30 by 11.3*3 F^ is closed in re&). II. 3.5 Theorem. Let (X,%) be a semi-uniform space and let~~

~~Bt=L Then B*£ T(U) if and only if 3 U cbi with U[B]=B.~~

~~P ro o f; Suppose th a t B+£ *¥$.)• Then 3 U eW with H(U)[B]£B"t.~~

~~In particular, U[b]£ S'*" so that U[b]sB.~~

~~Now suppose that 3 with U(B] = B and let A~~

~~C 6H(U)[A], CSU[A]c u Cb] = B. Thus H(U)[A] c b + and~~

~~II. 3.6 Corollary. Let (X/U) be a semi-uniform space; let~~

~~Be X w ith B"*~t *¥($£). Then B is open and closed in~~

~~Proof; By 11.3.5 3U<£^such that U[B] = B. By 1.1.29 k(B)=B and so B is closed. For any beB U[b]— B and so B i s open.~~

~~II. 3.7 Theorem. Let (X,%) be a semi-uniform space and let i:X ^ b y i(x ) =[xj. Then i:(X,^(ii))->(i[.X], i[xj i s a homeomorphism. Proofs The map is (X, ^(SOJ-^CijXJ* ^(iCx]xi[X] [\$L )) i s a~~

~~homeomorphism since i :(X,%)—^ i[x j, i[x]«i[x] £\°U.) is a unimorphism.~~

~~By 1.2.13 i[x] tftf(ijXI*i£xjA&) so that it is sufficient to~~

~~show ^(irx>iJ[x]A & )Gi[Xl/)"?(&)• Let tf£T(i[x]xi[x]n$L), l e t~~

~~■fx^ £ (X and l e t 0 = i -/ {(I). Then { x 'f £ Cl f o r ev ery x € 0 so that~~

~~{ x ^ i [ x ] / 10'c ^ , Thus ^£i[x]~~

~~II. 3.8 Corollary. Let X be a setj let ^ and V be semi~~

~~uniformities for X and assume that \ (& )= • Then $ 0 — ^ CV).~~

~~Proofs Using 11.3*7* it is clear that the identity map is a~~

~~homeomorphism from (X, (U) ) to (X, °C (V)) •~~

~~The next few results extend certain results of Caufield [ 3^ t °~~

~~the semi-uniform case.~~

~~II. 3*9 Theorem. Let (XjlL) be a semi-uniform space with~~

~~property K$ let i:X-*X by i(x)=fx}. Then i[xj is closed in (X3^(U ))~~

~~if and only if (X ,^^)) is T^.~~

~~Proofs Suppose that i[x] is closed and let x^y in X. Then~~

~~£x,y} ^i[x]:= k(i[x]) so that 3 U=U"‘ e ^ such that~~

~~H(U)[{x,yyni[xJ= p'. If zeU[xa^UCyl, then fx,y1sU£zJ so that~~

~~{z|e. H(U) j{x,yjj. Thus U£x] AUfyJ—jrf and, since (X,%) has property K,~~

~~(X, W ) i s t2.~~

~~Now suppose that (X ,^^)) is T 2. Let B£lc(iiX])« Then B~~

~~since H(U)[^Jn i[ x ] = jzf fo r a n y U e.7 1 . Thus |b \^ - 1 . Assume x ^ y and~~

~~^ x ,j^ £ B. There is U = U"'e^(. such that U£x]A U£yJ= j^. Let~~

~~{z\e H(U)[B] A itX]. Then-fx,yj£U£z] and so z gV[x] {\U[y3 which i s a~~

~~contradiction. Thus / B j ^ l so th a t Be i [ x ] . By 1 .1.30 i[x] i s~~

~~c lo se d . It is interesting to note that, if (X/U) is a semi-uniform space with property If, then £uju£ sU.'lz)—& if and only if (X,^(%.)) is T^®~~

~~In particular, 11.3*9 is the best characterization known to me of~~

~~Tg-spaces among the class of semi-uniform spaces with property K.~~

~~II. 3® 10 Theorem. Let (X/U) be a semi-uniform space. Then~~

~~(X,*r$)) is discrete if and only if (X,^ is discrete.~~

~~P ro o f; I f (X/U) is discrete, then .(X,^) is discrete so that~~

~~(^■3 °t(^l)) is discrete.~~

~~Now suppose that (X ,^^)) is discrete. Then [x], C ^ ($ .) so th a t~~

~~3 U fi-2iwith H(U) [x] = fxf. Let (x,y)eU and suppose x^y. Let~~

~~X-X-^j}. Then A<=U[x] and X£U[A] so that AeH(U)[x] which is a contradiction. Thus U = A and (X/U) is discrete.~~

~~II. 3.H Theorem. Let (X/U.) be ? semi-uniform space. Then the following are equivalent;~~

~~a) ($,W )) is Tq~~

~~b) (Xt^FCU)) is discrete~~

~~c) (X,^(&)) is T2.~~

~~Proof; (at^>b) Let AeX. Since k(A)£U[A} for every U €% , i t is clear that (A,k(A))eH(U) for every U c i l . Thus, for any~~

~~0 G ^U ), Aed if and only if k(A)c6. Since (X^OtL)) is T0, the points A and k(A) cannot be distinct, that is, k(A)=A. Thus~~

~~is discrete.~~

~~( b ^ c ) L et A-^B in X. Case I ; There i s x&A with xj^B. Since~~

~~(X,¥(i()) is discrete, there is U=U“*£% with UCxJ^xl* Then~~

~~=P{xj. By II.3.3 and 11.3.5 i x l ~ e ‘fltU ) and~~

~~(Cfx?)+£ ^ 6 ). 1»2~~

~~It is clear that k e i ^ Y ) B a n d fx 3 f\ ( £{x})*~~~

~~Case II; There is x£B with x^A. Then A and B can be separated exactly as in Case I.~~

~~(c=^a) This always holds.~~

~~II. 3.12 Theorem. Let be a semi-uniform space with property K. Then ($,^(&)) is stable if and only if for every~~

~~A c t 6(£a?)={bJc(B)=c(A)].~~

~~Proofs Suppose that (X, '£''(&)) is stable and let A e X. By~~

II.2.1i{B|c(B) = c(A)3 = k(fAj)oc({A]). Let B£c( {A]). Suppose c(A)^c(B). Then A £ ( £c(B )) s o that c(£a^)£ (£ c(B)) • But then BiOCc(B)^ which is a contradiction. Thus c(A)£c(B). Using the fact that Ae c({b!) (1.1.20) and a similar argument, c(B)£c(A),

~~Now suppose that c^ A ^ ^ - ^ b J c(B)= c(A)^ for every AeX. Using~~

~~II.2.U and 1.1.31, c(fAl):s k (W )£ fl{8)A6 6 Thus (X,*F$)) i s s ta b le .~~

II. 3*13 Example. Let X be infinite and let 'V be the co- finite topology for X. Then (X, i(£t)) is a semi-uniform space with property K. As in II.2.£, 1c(^(fx])) consists of all non-empty subsets of X. Thus for any xe X,{x^£ $(fx$). By II.3.12 (X,TC )) is not stable.

~~It is interesting to note that, using a result to be proved in the next chapter (IH .6.7), (X,cf(^l^ :))) as in II.3.13 is compact.~~

~~This gives an example of a compact semi-uniformizable topology which is not stable.~~

~~The last two results in this section extend certain results of~~

~~M ichael [ 13] • II. 3e111 Theoremo Let (XjU.) be an HK space and let % be closed and compact in (X ,^^))<> Let FQ= l/^F |f and F is closed^.~~

~~Then FQ is closed.~~

~~Proof; Let xec(FQ) and suppose x ^Fq. For each F c°K w ith F clo sed 3 UF £ ^ with UF{;x]AF = jzf and 3 VpsVjT'eU with~~

~~VpoVpfxJ G. Up[xJ, Suppose ye Vp[xjf\ vf [f]» Then 3 zeF such that~~

~~(z,y)£ Vp, Then z£VFoVF[x] so that UptxJHF^^ which is a contradiction. Thus 7F[x]A ^[?l=/* Let B £%. Since % is closed, c({B})<^. Then c(B)e%' and B€lnt H(V c^b j )[ c(B)3 . Thus~~

~~% V { i n t H(Vp )[F ]j F £ % and F is closed^. By compactness of %~~

~~3 F-^,.»e,Fn£°}t with each F^ closed such that~~

I n t H O ^ I fJ. Let V=/\ Vp and let yeVM AF^ Since y£F0, there is F with F closed and yeF. There is F.j_ such that F€ H(V )[F j. Then ye-F£VF.[F3 and ye V£x]G. VF.fxJ so that x i * x VF£ x] f\ VF> [ F j ^ <}>, a contradiction.

~~II. 3«l£ Theorem. Let (X/W) be an HK space and let ^ be a family of compact subsets of X; assume that % is compact. Let~~

~~V = 1/{k|k£^. Then Kq is compact.~~

~~Proof: Let-|'oti; pie&^ be an open cover of KQ. For each~~

~~K e % 3 a finite set A j^A such that KS U£0^ |cxe A ^ 0 For each xeK 3 Ux such that Ux<>Ux[x]<£. u£o<* W e^Kl* There is a finite set FkSK such that K £ U{ux[x] j x£ F^. Let %=^ { u j x eF ?. Then~~

~~% £= U{h(U^)[k] |k £ 9 ^ and this a neighborhood cover. By compactness~~

~~3 K-j_>... jKn £ such that ) [.KjJ. Let x&Kq. There is~~

~~K c°)i such that xeK and 3 Kj such that Ke. H(H^)[Kj]. Then~~

~~3 x1 £ Kj such that (x^xJgUjj and 3 xM£F^ such that 3 J ,x')eux,,. Then xe ux , ,oUx , , [ x " J c \J [ o * j a e A ^ ] .~~

~~= l j • • • j n^ « CHAPTER I I I~~

~~THE HYPERSPACE OF A TOPOLOGICAL SPACE~~

~~III. 1. Definition And General Facts.~~

~~III. 1.1 Definition. Let (X/£) be a topological space.~~

~~C D ) and *£= *?( 2$*)). The space ($,£) is the hyperspace of (Xff).~~

~~As might be expected, a topological space and its hyperspace are most closely connected when the original space is stable. Results throughout this section w ill make this precise.~~

~~III. 1.2 Theorem. Let (X/t) be a topological space and let i:X—*X by i(x)=-fx|. Then i:(X #)-*(X /J).~~

~~Proof: By II.3.7 i: (X ,^*)->(i[x],i[x]f)r) so that i : ( X, I f ) . Then i : (XfZ)-* (X/t) since .~~

~~IU . 1.3 Theorem. Let (X/£) be a topological space and let isX-^X by i(x)= fxl. Then (X/fc) is stable if and only if i:(X ,^)->(iL x],i[x]nr) is a homeomorphism.~~

~~Proof: By II.3.7 i:(X ,^*)-*(i [x],i[x]/) £ ) is a homeoraorphism.~~

~~Thus i: (X/£)—*(i[X j,i[xJn/£') is a homeomorphism if and only if~~

~~°Z'*~eC>, By definition (X,^ is stable if and only if .~~

~~III. 1.1: Theorem. Let X be a set and let ^ and be to p o lo g ie s f o r X. Assume t h a t "2^ ^ Then <7t^ A.~~

~~Proof: This is immediate from II»3»8.~~

~~I I I . 1*5 Corollary. Let X be a set and let and ^ be stable topologies for X. Then T, =• if and only if ^ ^ .~~

~~Proof? If ^ clearly ^ = %. • If ^ , by HI.1.1;~~

~~The next example shows that, given (Y,ct'/)£(X/fc'), (Y,0^'') need~~

~~A _A not be a subspace of (X/c). The results which follow give some partial results about subspace relations.~~

~~III. 1.6 Example. Let X = [o ,l] and l e t~~

~~Jojo£[o,l) or 0 = [ o , l ] ^ Then U (r^={xxxj. Let Y= [0 ,1 ).~~

~~Then (Y,Y/l‘rT) is discrete so that (Y, %(YA^)) is discrete. Thus~~

~~(Y,YAT) is discrete. Clearly, (YjYA'V ) is not a subspace of~~

~~( x /£ ).~~

~~H I. 1.7 Lemma. Let (Y ,^)^ (X/c"). Then YxY/1 IL~~

~~Proof? Let U e.% & ) and let y eY. _ Then 3 0 £ ^ such that~~

~~(y ,y )£ O x O £ U . Then (y ,y ) £ (0 0 Y)*(OAY)£. Y*YAU. Thus~~

Y*YAU eU(Y')*

~~III. 1.8 Theorem. ■ Let (Y,*fc")£(X/£). Then Y~~

~~Proof? Since YftY/W.(ct)^ .‘U(^},/)i it is clear that~~

~~YVYVuTc^). Thus ^(Y^Y/vSfc))£*?'. By II.1.7~~

~~YxY/V?fc)^ YxYA'l^) and by 1.2.13 YA^ Y x Y .~~

~~The next definition follows Levine [11 J .~~

~~H I. 1.9 Definition. Let (Xf£) be a topological space and let~~

~~A£X. A is generalized closed if and only if A^O with 0 implies c(A)£0.~~

~~III. 1.10 Lemma. Let (Y/T')<^(XfZ) and let Y be generalized c lo sed in X. Then U (Y )~ Y*Y A W ) .~~

~~Proof? By III.1.7 Y^YAtU(cV)S‘k(/^/). Let U £^(V ). For each y e Y 3 0*£ ^ /such that (y,y)<£ 0 0 ° <^U and 3 0 such that y o~~

~~Ve U(%). Let (y^y^S YxY AV. By the definition of O ^ypy^e W. f | Thus 3 y e Y such that (y-^yg) £ °y ®y° Hence (ypyg) £ OysOy^U and so U £YxY A'UCAr).~~

~~UIo 1.11 Theorem. Let (Y /t')£(X /t) and let Y be closed in~~

~~(X/T*). Then (£/?') ^(X /£).~~

~~P ro o f: By I I I . 1.10 (Y,U Cfc"))S(X,% (X)) and so by n .1 .7~~

~~(Y, 2^'))£(X ,% (V )). Clearly Y=Y'*'and so by II.3.U Y is closed in~~

~~(X,£). By 1.2.15 (Y/?)^(lft).~~

~~III. 1.12 Theorem. Let (Yft')~~

~~Proof: By in .1.10 (Y,% C£'))£(X, W f)) and so by II.1.7~~

~~(Y,'%Ct’/))£ (X, %($£)), Since (X,^(^)) has property K, by 1.2.16~~

~~A A A* a (Y,V)£(X,'t. The remainder of this section w ill deal with continuous maps and the natural hyperspace map.•~~

~~III. 1.13 Lemma. Let (X,^) and (Y/TJ be topological spaces and let f^X ^-K Y ,^). Then f:(X,2*(^))-*(Y,ft {%))•~~

~~Proof: Let ’7^-tU (X 3) and let x£.X« There is an Og c °VX such that (f(x),f(x)) £0gX0g£Vo Then~~

~~(x ,x )G f",[02]xf~,£0g] £ (f* f )~l[ Ogx 0g}=. (f^f r ^ £ v ] . Thus, since f"‘[0g}e^ , (fxf)"‘C v ]e^ (0 ',).~~

~~III. l.ll; Theorem. Let (X ,^) and (Y/% ) b® topological spaces Ii8 and let f:(X,?;)-*(Y/fc;). Then f*(X,^*)-»(Y,^*).~~

~~Proof; By III. 1.13 f :(X, % {ft, ))-»(Y,3t (0^)) and so, by 1.1.15, fs(x,^,*H>(Y,r*).~~

~~III. 1.15 Theorem. Let (X,^Y;) and (Y,45^) be topological spaces and let f ; (X ,^)—»(Y,°%). Then f : (X,<£;)-*(Y ,^).~~

Proof; By III.1.13 f:(X, ^((% ))-*(Y, *£(*%)) so that f:(X,W r() H ^ (^ ) ). Thus £s(X,£,HK£#*)• III. 1.16 Corollary. Let (X,^) and (Ys%) be topological spaces and let f ; (X, )-^( Y, ) be a homeomorphism. Then f:(X, )->(Y, is a homeomorphism.

~~Proofs By II.1.13 f is 1-1 and onto and f~‘= f ”1. By the last~~

A A » theorem both f and f ' are continuous*

~~III. 1.17 Example. Let X«{a,b?, let ^ = ^ ,{ a ] ,X^ and let~~

~~^sJV ^b^x}. Then U '% )~K. (^J= ix*xj. Let I:X-»X be the identity map. Clearly I; (X,U (ft, ))-> (X ,^ (% .)) so th a t~~

~~I ; ( X, 2^V,)HKX,' ) . Thus I: (X,£,)->(X,4;) while I is not continuous from (X,%) to (X/%).~~

~~III. 1.18 Theorem. Let (X,5^,) and (Y,*%) be topological space &} l e t f : X-*Y w ith f : (X, % Yy% ) . Then f : ( X, %* )-X Y, ^ * ) .~~

Proof: Let ix:X-»X by ix(x) = {xl and let i y:Y-*Y by iY(y) = £yk*

~~C le a rly f = i~ ‘ofoir By I I . 3 . it i x # (X,%*)-*( ix[x],i^jx] ) and iY:(X ,^)-^(iYW ,iyW ) are home omor phi sms. Thus f*CX,<^)-K Y ,^ ).~~

~~III. 1.19 Theorem. Let (X ,^) and (Y,°c() be topological spaces with (Y,*%) stable. Let f:X-»Y. Then fs(X ,^)->(Y ,^) if and only~~

~~A A A , A A v if f:(X ,^)-»(Y ,^). Proof; If f:(X,=r,H>(Y,rJ, by H I.l.l£ f : (X/f, HKY,#J. If~~

~~f:(X,c*r/)-^(Y,‘5£;), by I II .1.18 f:(X, *£?)-*( Y,*r*). Then, since~~

~~^ * = % and % 3 .~~

~~III. 1.20 Corollary. Let (X,0^) and (Y,^) be stable topologi cal spaces and let f:X-»Y. Then f;(X3% )-*(Y ,^) is a homeomorphism~~

~~if and only if f ; (X/^ )^(Y ,^) is a homeomorphism.~~

~~Proof; By III.1.16 if f ;(X,‘r^’)-^(Y,‘r^) is a homeomorphism, then~~

~~Pi(X,^)— is a homeomorphism. If f ; (X3%)—HY,^) is a homeomorphism, then f is 1-1 and onto. Since (Xf%) and (Y ,^) are~~

~~stable and III.1.19 holds, f :(X,%)—KY ,^) is a homeomorphism.~~

~~The next few results characterize stable spaces in terms of various mapping properties.~~

~~III. 1.21 Theorem. Let (Xf?) be a to p o lo g ic a l sp ace. Then~~

~~(X/£) is stable if and only if for every space (Y ,^) and f:Y-»X, f:(Y, KXj^C^)) is equivalent to f;(Y ,^#)—^(X/^).~~

~~Proof: (Necessity) Let (Y,^) and f:Y-»X be given. If f:(Y,2/(^))-^(X,^(^)), then f: ( Y/Tf )-»(X/*"*). Since ,~~

~~Pi(y,‘T/)-»(X ,^). IP fs(Y3^ ‘>r)-^’(X/T)j using f;(Y,‘ri;)-*(X,cV). By III.1.13 P:(Y ,^(^))-^(X ,^(^)).~~

~~(Sufficiency) Let I:X—*X be the identity map. Then~~

~~I : ( X, (ft))—>(X, (21) ) so that I ;(X/V*H>(X/b). Thus~~

~~III. 1.22 Theorem. Let (X/t) be a topological space. Then~~

~~(X/b) is stable if and only if for every space (Y,%) and f ; Y—*X, f s ( Y is equivalent to f :(Y/%)—>(X/£).~~

~~Proof: (Necessity) Let (Y, %) and f:Y—»X be given. If~~

~~P:(V*?)-KX/)r), using f:(V *'/)-*(X/*) and so f:(Y,*r; )-5>(X/*0. If fsC Y ,^)-^/?), then by III. 1.18~~

~~Since ^ = f : (Y^H^X/?).~~

~~(Sufficiency) Let I:X-*X be the identity map. Then~~

~~I:(X/9)—?(X/^) so that I:(X/T^-HX,^). Thus cT ^ rc .~~

~~III. 1.23 Theorem. Let (X/?) be a topological space. Then~~

~~(X/V) is stable if and only if for every semi-uniform space (Y,V) and f:Y-*X, f : (Y/KH^X,^ (V)) implies f^Y ^CV ^^K X ;^).~~

~~Proof; (Necessity) Let (Yj'y) and f ; (Y,‘V)-KX,% f^)) be given.~~

~~Then f;(Y,*T(V)) and ‘t'^ T s o that f;(Y,^(V))->(X,^).~~

~~(Sufficiency) Let IsX—?X be the identity map. Then~~

~~I : (X,$L (*t) h * ( X, W & ) ) and so I : (X/*r* )-* ( X # ) . Thus ^ ^ "T*.~~

~~III. 1.2ii Theorem. Let (X/£) be a topological space. Then~~

~~(X/£) is stable if and only if for every semi-uniform space (Y,V) and f :Y—=»X, £:(Y#)~»(X,9f&:)) implies f:(Y,*fcW )-*(X,/V).~~

~~Proof; This follows immediately from III.1.23 since for any (Y,V) and f:Y-2X, f ; (Y/V)-»(X,t< (V)) if and only if f:(Y ,V >-»(X ,^r)).~~

~~The next few results concern the hyperspace of the weak topology induced by a map. In particular III.1.32 is a better result than might have been expected at first glance.~~

III. 1.25 Lemma. Let X be a set, let (Y/'fc) be a topological space and let f:X~*Y. Then (fxf)“' (*

Proof: Let (Z ,^ ) be a topological space and let g:Z-*X*X. It is sufficient to show that g:(Z ,^ )-*(X*X, f'^ x f -*^) if and only i f (fx f )o g: ( Z, )—>( YxY, * tx 'V ) . Since f*f:(X*X, f"'rx )-*(Y*Y, r* r), if g is continuous, then (f*f)*g is continuous. Now suppose that (fxf)a>g is continuous. Let IT, and projections from Y*Y to Y. Then f °7ft o g = p-j£> (fxf)og which is continuous and so 71, o g is continuous®

~~Similarly, JTX o g is continuous. Thus g is continuous.~~

~~III. 1.26 Lemma. Let X be a set and let (Y/£) be a topological~~

~~space; let f sX—*Y be onto. Let U be open in f“/(Vx f "• *2? and l e t 17*= (fx f ) [u ]. Then I^e^ C 'V ) and ( f x f r'[u*]:=tj.~~

~~Proof; By III.1.2^ 3 V etxT such that (fxf)"' [vJ=U. Since f is onto, V s (fxf) [>(fxf)",[v]]=(fxf)[u]sU^o Thus (fxfU and U* d Moreover, since f is onto and AX£U, £ U*. Thus~~

~~U * e % fV ).~~

~~III. 1.27 Theorem. Let X be a set and let (Y/t) be a topological space. Let f:X-*Y be onto. Then )=f ~l ^(V),~~

~~Proof: Since f; (X,f"'^')-^(Y,^), f : (X, °& (f‘"t*))-*(Y, U (V ) •~~

~~Thus f 1 °U^t) £ % ( f~,aY ). Let U £ ^ (f-^ ) and assume without loss of generality that U is open in f-,c^x fT iDt « By III.1.26~~

~~U* = ( fXf )[u l £ °U (?) and U = (fxf)' 1 [u * ] . Thus Uef‘'^(^).~~

~~III. 1.28 Corollary. Let X be a set and let (Y/£) be a topological space; le t f :.X—>Y be onto. Then~~

~~Proof: By III.1.27 By II.1.17~~

~~° u x r ^ t ) = = f'm* t f d k ). III. 1.29 Corollary. Let X be a set and let (Y/£f) be a topological space; let f :X-»Y beonto. Then f ~ ‘°Z —~~

~~P ro o f: By III.1 . 2 8 ) = f ”i tl 0?), Since $ is onto, by~~

~~1.2.29 f^V = f‘'*‘^ .~~

III. 1.30 Example. Let X“ £a,b| and let Y = £a,b,cJ. Let ^ ,{ a } ,{b] ,^a,b} ,Yj; and let fsX-^Y be inclusion. Then f x''C ) i s discrete. Thus £~isX is discrete. On the other hand, U (V) Y Y so t h a t f °tl {V) ■£■ U(£~'it ) . A lso' *£ = [$, f&j, y}. Clearly,

~~r ^ t and f o l ] £ f"’^ . Thus f^V ^ f “/‘rr.~~

~~III. 1.31 Lemma. Let X be a set and let (Y/t') be a topological spacej let fsX-^Y. Then f~ ,Dt-= f"' (f[x]/lQT).~~

~~Proofs This follows immediately from the fact that, for any~~

~~A*S-Y, f " ‘[A ]= f " 1 J f JfxJn a J .~~

~~III. 1.32 Theorem. Let X be a set and let (Y,‘D be a topologi c a l space; l e t fsX-»Y. Assume t h a t ( f [ x j , f[X]/)1^ )£=(Y,T). Then~~

~~Proofs By III.I. 3 I f’V - f“;(f/x ]^ ). Then, using III. 1.29, fT ^ t = r^TfpGA^ )- f(f[x ]/n r). Since flx]^r= ffxjnr , by h i. 1.31 £ ~ ^ - r '(ffxjA^J^f"'^.~~

~~III. 1.33 Theorem. Let (X,^) and (Y ,^) be topological spaces * A A with (Y,1^) stable. Let fsX—»»Y and assume ^ = f • Then~~

~~% * = £ - '% .~~

Proofs Let (Z,5^) be a topological space and let g*Z-*X. It is sufficient to show gs (Z,^,)—^(X,Qr,) if and only if f«gs(Z/XHKY/X). By III.1.18 fs(X,^y-*(Y/V£) and so fsCX,^)—^(Y,^). Thus, if g is continuous, fog is continuous. If fog is continuous, then fCgsfogsSince

~~= gs(Z/&HKX,'&). By IH .1.18, gs(z,r/H>(X,^) and, sin ce *¥*—%>, gs(Z,^)~^(X,^).~~

~~III. 1.3U Example. Let (X,^) be [0,lJ with the usual topology f>3~~

~~and let Y =|a,bj. Let f:X->Y by f(x) = a if xejp,^] and f(x)=b if~~

~~The q u o tie n t topology f o r Y i s . Then~~

~~‘U( c¥') = {Y*yJ and so Y^. Also £ c o n s is ts~~

~~of all non-empty subsets of £o,§J which is closed in ^ . Thus is~~

~~A not the quotient topology for Y. ... III. 1.3£ Theorem. Let (X,1^) and (Y, *%) be topological~~

~~spaces with (Y,.*^) stable; le t fsX—*>Y and assume that f :(X/T,)—^(Y,^,.) is an identification. Then ^ is the largest~~

~~stable topology for Y such that f : (X /^J-^Y ,^).~~

~~Proof: Since (Y,f£) is stable, by III.1.19 f :(%,% )-*(Y/T3).~~

~~Let ^ be a stable topology for Y such that f^X ^J-^Y ,^). Let~~

~~I:Y-»Y be the identity map. Then I«f K X ,^)—^(Y,^) s o th a t~~

~~I^fsCX,^)-^^,0^). Then, since I°f= i°f and f is an identification,~~

~~I:(Y,£)-*(Y/%).~~

~~Thus ^ .~~

~~III. 2. Comparisons With The Finite Topology.~~

~~III. 2.1 Definition. Let (X/f) be a topological space. is the topology for X with subbase { 0~}o £ ‘Yj* is the topology for~~

~~X with base { 0+j 0 £ The finite or Vietoris topology is ‘V V~~

~~H I. 2.2 Theorem. Let (X/£) be a topological space and let~~

~~A^X. Then A_e ^ if and only if A e '^*.~~

~~Proof: This is immediate from II.3.2.~~

~~III. 2.3 Corollary. Let (X/f) be a topological space. Then~~

~~^ ^ ^ if and only if (X/£) is stable.~~

~~Proof: If ^ then, by H I.2.2, 0~~

~~(X/2T) is stable. If (X,^ is stable, then and so . 51» III. 2.1; Theorem® Let (X/'zf) be a topological space and let~~

~~AS. X, Then A+ e <7t* if and only if A is open and closed in ^ ®~~

~~Proofs Suppose that A+e. By II®3<>3 3 U £ ‘UffZ) such that~~

~~U/A] = A® Thus A is open and closed in ^ . Suppose now that A is~~

~~open and clo sed in L et U=A*A U Ck * £ k , Then U and~~

~~U[A}=A so that, by 1 1 . 3 . 5 , A+ t # .~~

~~Ill® 2.£ Corollary® Let (X/£) be a topological space. Then sy+c; && 1 — ^ if and only if every open set is closed.~~

~~Proofs This is clear from III.2.1u~~

~~III. 2.6 Definition® Let (X/£) be a topological space. ^ is~~

~~the set of closed sets and C(X) = |A~~

~~III. 2.7 Example. Let (X/f) be [0,1^ with the usual topology.~~

~~Let (% ,3]* ,lj. Since ‘UC’iT) is a uniformity,~~

~~H(U)£|oj] is a ^-neighborhood of ^0^. Suppose 3 0^,...,0n£ such~~

~~' t h a t { 0} £ A ojc H(U)[£oi] . Then {0,l} £ f\0“. But {o,l]^u[oQ~~

~~which is a contradiction. Thus H(U)J|ojJ is not a ^-neighborhood of~~

~~{ 0} • This example shows that, in a stable space, in general atf'j£e£‘.~~

~~Moreover, ^ ^ and C(X)AeT"^ C(X)A'?' .~~

~~III. 2.8 Example. Let (K/t) be the reals with the usual~~

~~topology and let P be the positive integers. Let~~

~~Vi=£(x,y)l |x-y|~~

~~H (7i)[pJ i s a ^-neighborhood of P. Suppose 3 0Q,0 p . . .,0 n <=. such~~

~~that Pe0Qf)0jA...A0“£H(V1)[pJ . (Note that P tO ^ H ^ fp ] is~~

~~impossible since jrfeO ’*’. Also, if P£ 0^ f \... A0~<=.H(V-^)[p], then~~

~~P€ XÂO^/)... A0“<=:H(V^) P .) For each i^ l, let kêOÂP. Let~~

~~B -^kp...,kn^. Then B£P^0Q and BAO^^1^ for each i^ l. Thus B 6 o* AO£A... AO^ and so PSV^Bj . Since B is finite, this is a contradiction. Thus H(V^)[P^ is not a V^^-neighborhood of P.~~

~~This example shows that, even if (X/£) is stable, in general~~

~~III. 2.9 Theorem. Let (X/^T) be a topological space. Then~~

~~C(X) A ^ £C (X )A f T ~v~~

~~Proof? Let() 6 *c and le t A £ O AC(X). If~~

~~A =^, A£ jTAC(X)£OAC(X). Assume A=jrf. Since A £ 0 ,3 ^ ^ W such that H(U)JaJGO. For each xeX, 3 0X£ <^ such that~~

~~(x,x)£ 0^ 0X~~

~~B ^ 0 X>^ U[a 1 and, sin ce B A O ^ ^ /^ f o r each i , A —JJ, U[b] .~~

~~Thus B£C(X)A0 and C(X)AO£C(X)A(/V "v‘t +), Note that, taking~~

~~(X/T) compact and connected, C(X)=X and by III.2.5 ^ ^ ^~~

~~III. 2.10 Example. Let X be infinite and let ^ be the co- finite topology, Let x 0 6 X and let 0 ” ^x js . Suppose that~~

~~Â0+ £ Â ^. Then 3 S such that = ÂO"*". Let x £ CfxJ,. Then jx3 £ Â 0 and so 3 U such that~~

~~H(U) [{x?]c[S. Thus U[x3e0. (Note also that U [x]£^.) There exists Ve U(*t)3 H(V)[ucx]]c 6 . Since (X#) is compact,~~

~~3 0^, .••,0n£^/ such that each 0^^ and For each i, since bothÔ^ and £(U[x]) are finite, 3 xêOÂU£x]. Let~~

~~F = ^xq,x^, ...jX ^ . Then U[x]<£V£f]— X and~~

~~F&vfULx]]=X so that Fe % /\0. . But F d0+ which is a contradiction. Thus /) ‘cfr'Y?.~~

~~In the next lemma the following fact w ill be used: If (X^JT’) is~~

~~normal and ^Un~ ^ U ) ^ a finite open cover 0^, . ..,0 n with~~

~~& C L ^ U ^ x ], then 2^ is a uniformity for X.~~

~~III. 2.11 Lemma. Let (X/f) be normal and let it* be as above.~~

~~L et U£ li* and let AaX. Then H(U)[a] is a ^-neighborhood of A.~~

~~Proof: , and so &(V). Thus 'V(#*)£c’£'. Since~~

~~is a uniformity, H(U)[ a 3 is a -neighborhood of A. Thus~~

~~H(U)[a J is a ^-neighborhood of A.~~

~~H I. 2.12 Theorem. Let (X/£) be normal. Then ^ / ) % f\ *V.~~

~~Proof: Let F £ °&C\ O'4' for some 0 . Then F£0. Let~~

~~U =0*0 UCF*Cf . Let F* € H(U)[f]. Then f'gu(f]=0 and so~~

~~F 1 e *£/) O'*. By III.2.11 H(U)[f] is a *T -neighborhood of F. Thus~~

~~0+ £~~

~~III. 2.13 Corollary. Let (X/t) be normal and stable. Then~~

~~Proof: By III.2.3 and by III.2.12~~

~~. Thus % A(f v'O* (^r)y ^ n ' f .~~

~~III. 2.1U Theorem. Let (X/£) be compact and regular. Then~~

~~Proof: Since X is compact, C(X)=X. Thus by III.2 .9~~

~~and so ^A 'fe ^ A (^"V ^+)» By III.2.13~~

~~III. 3. HK Spaces.~~

~~III. 3.1 Definition. Let (X/ir) be a topological space. (X,Ar)~~

~~is an HK space if and only if (X,ii(A*)) is an HK space. III. 3.2 Theorem. Let (X/2r0 be an HK space and let Y£X be generalized closed. Then (I,Y /3^) is an HK space.~~

~~Proof; By III.1.10 %(Y/VV)=YxYO‘U(fiW« By II.1.7~~

~~(£tf?Y ^))£(X ,4£cb) and so (Y, ^Y nV )) has property K. Thus~~

~~(Y,Y A^V) i s an HK sp ace.~~

~~III. 3»3 Theorem. Let (Y/c) be a topological space; let X be a set and let f;X-»Y be onto. Then (Xjf*1^ ) is an HK space if and only if (Y/£) is an HK space.~~

~~Proof; By III.1.28 2Z(f~*V)=f Then by 1 . 2.30~~

~~(X ,ii(f”V )) bas property K if and only if (Y, °U.(?0) has property~~

~~K. Thus (X ,f"'^) is an HK space if and only if (Y /if) is an HK space.~~

~~III. 3.U Corollary. Let (x/^ ) and (Y ,^) be homeomorphic topological spaces. Then (X/^) is an HK space if and only if~~

~~(Y,5^) is an HK space.~~

~~Proof; Let f ;(X ,^)->(Y ,^) be a homeomorphism. Then~~

~~'Y, —f'"*^ and so the conclusion follows from III.3.3•~~

~~III. 3*5 Corollary. Let (Y/k) be an HK space; let X be a set and let f;X-*Y. Assume that f[x] is generalized closed. Then~~

~~(X ,f*^) is an HK space.~~

~~P ro o f; By I I I . 1.31 f “**£ « fH (f£x] Regarding f as a map onto f [X], by III. 3.2 and III.3.3 (X, f “'(f [x] /Vfc)) is an HK space.~~

~~Thus (X ,f~V ) is an HK space.~~

~~III. 3*6 Theorem. Let (X/V) be an HK space. Let and Fg be closed in ^ with F^A Then 3 OpOg e su°b that~~

Fl - ° 1> F2-°2 and 0lA0g=^. Proof; Let U = &F-j* t F^UCFg*£Fg. Clearly U e ^ . By I I . 2.12 3 VaUft) such that VoVj>J ^ U jlJ. Since (X, l i f t ) ) has property K, there exists an 0'^ £ with ° Then c*(0*)£v[0*]^VoV[F1] ^ U[Fj = £ F 2 o Let 0*= £c*(0^). Then ofe^FgSO g and 0^0*=/*.

~~III. 3»7 Corollary. Let (X/£r) be an HK space. Then (1ft) i s norm al.~~

~~Proof: This follows from 111.3.6 since c S c .~~

~~III. 3«8 Corollary. Let (X/£) be an HK space. Then ( l/tf) i s stable and normal (and hence completely regular).~~

~~Proof: (X,^*1) is stable since (X, &CV)) has property K. Since~~

~~°X ^<=.0Z , IH .3»6 immediately yields the normality of (X,^*).~~

I I I . 3.9 Example. Let (1ft) be a compact, Tg topological space in which 3 Y£X with (Y,I) not normal. (Such spaces exist: for example, the Tychonoff plank.) Then (1ft) is an HK space since %Lft) is a uniformity. By III.3.7 (Y,Y/I^) is not an HK space. Thus, in general, a subspace of an HK space need not be an HK space.

~~III. 3»10 Example. Let (1ft) be the half open interval space.~~

~~Then (X,^) is paracompact and regular and so ( l , l l f t ) ) is a uniform space. Thus (1ft) is an HK space. However, (X*X,) i s n o t normal. Thus the product of HK spaces need not be an HK space.~~

~~III. 3-11 Lemma. Let (1ft) be a topological space. Then (1ft) is normal if and only if, for every U £ %L( ) and A£X, 3 V £°UL ft) such th a t VoV[A'J £ u [a] .~~

Proof: (Sufficiency) Since V[xj is a “neighborhood of x for every x eX and V £*}!(*¥) and c(A)SV[aJ for every AS.X and V eflL (*t), the proof used in 111.3.6 works here. (Necessity) Let A£X and let U Assume without loss of generality that U cl llYxpYa Then £*¥ and c (A )S U [a J. By normality, £[ 0-]_,0g £ ^ such that c(A)£O^C c(0- l ) ^ 02^ c ( 02)£ U [X ].

~~Let 0y= U/Al-c(01) and le t V=0g*02U O-jXO^U~~

~~V eU (c) and V[A]=Og. Also v[b^£0 2 UC>3 = U[a[] and so~~

~~VoV[A]£U[A].~~

~~IH . 3.12 Theorem. Let (X/c) be a topological space. Then~~

~~(X/£) is an HK space if and only if (X/£) is normal and, for every~~

~~A <£X and A(U) is a ^-neighborhood of A.~~

~~Proof: This follows immediately from II.2.12 and III.3.11.~~

~~I I I . lu Weak S e p aratio n P r o p e rtie s .~~

~~III. ii.l Theorem. Let (X/£) be a topological space. The following are equivalent:~~

~~a) (X,&) is T0~~

~~b) (X/£) is discrete~~

~~c) (X/£) is discrete~~

~~P ro o f: (ar> b) By I I . 3 .1 1 , sin ce (X ,£) i s T0, (X,*£*) i s discrete. Since (X/t) is discrete.~~

~~( b ^ c ) I f (X/t) is discrete, then (X ,^(^)) is discrete. Thus~~

~~(x,-u1 V)) is discrete and so (X,*^) is discrete.~~

~~(c=r>a) This is clear.~~

~~III. JU»2 Theorem. Let (Xf?) be a topological space and let i:X -?X by x(x)=z{x}. Then i[x ] is closed in °Y if and only if (X/^) i s Tg.~~

~~P ro o f: Assume t h a t (X /t) i s Tg. Then (X , *U(?Y)) has property K and ^ ( U ( X ) ) - ^ » By II.3.9 i[x] is closed in (X,$). 60~~

~~Now assume that i[x j is closed in and let xj=y in X. J u s t as in I I .3 .9 3 U eft.(T) such that V[xjf\U [y 3 = ^. Since U£xJ and U[y3 a re °£ -neighborhoods of x and y, (X,T) is Tg.~~

~~A A Since a subspace of a stable space is stable, clearly (X/T) A A stable implies (X,T*) stable. The next example shows that (X/t) stable need not imply that (X,^(T)) has property K.~~

~~III. Iu3 Example. Let X=£a,b,cJ and let 0 j 0 = ^ or b £ oj,~~

~~consists of a ll supersets of B = -|a,b}x'{a,b^U4b,cJx.|b,cjL As above, (X,i/l(T)) does not have property K. Let 0 £.°t w ith 07^ and~~

~~0 = Thus 3 AeO such that Then H(B)[aJ£0. Since A^/rf, b e B[A] and A^B[b] = X. Thus 0. Clearly H(B)^b]]-=£ftI and so 0-=X or 0 —Clfil* Thus "V— and (X,T) is stable.~~

~~III. ii.it Theorem,. Let (X,D be a topological space and assume~~

~~(X,ft(T)) has property K. Let c* be the closure in ^ . Then (X/t) is stable if and only if for every F closed in and BSF with c * (B )^ F 3 U eft ft) such that C£H(U)[b] implies c*(C)^F.~~

~~Proofs (Necessity) Let F be closed in and let B5LF with c ^ B ^ F . By I I .3 .1 2 B ^c({f}) and so 3 U eft (T) such that~~

~~H(U)[B]^Cc(^F». By II.3.12 CeH(U)/>] implies c*(C)^F.~~

(Sufficiency) Let A £ X. By II.3.12 and II.2.U it is sufficient to show that k(*fA3)=.jB|c' 3f'(B)= c^CA)^ is closed. Let Be X with c*(B)gfcc*(A). If B£ [j£c*(A)] which is open, 3 U eft(T) such that

~~If BGc*(A), by hypothesis~~

~~3 Ueft (T) such that H(U)[B]^£k({A?).~~

~~I I I . h»!i> Theorem. Let (X/D be a topological space and assume ^ » (X, V.ifX)) has property K. Then ,(X,T) is stable if and only if for 6 l~~

~~every F closed in and BSF with c*(B)=£F 3 TJ G-ltO?) such that~~

~~C^-U[bJ im p lies c 7r(C )^ F.~~

~~Proof; (Necessity) Let F be closed in and let B^F with~~

~~c*(B)£F. B ylll.ii.i; 3 U ei{(t) such that CaH(U)[B] implies~~

~~c*(C)?fcF. Let C£U[B] and suppose c*(C) = F. Then Bsc*(C)£U[cfl so~~

~~t h a t CeH(U)j^B] and c^(C)^ F which is a contradiction.~~

~~(Sufficiency) Let F be closed in V j let B^F with c*(B)^F~~

~~and l e t V&lKft) be as in the hypothesis,, Then C£H(U)[bJ implies~~

~~CSU[B] so that c*(C)=£F. By IH.U.l; (X/£) is stable.~~

~~Recall that in 11.3*13 an example is given of (X/^) stable with~~

~~A> -A (T/q not stable. The next theorem gives regularity as a sufficient~~

~~condition for hyper-stability; the example which follows shows that~~

~~regularity is not necessary.~~

~~I I I . ii.6 Theorem. Let (X/iO be a topological space; assume~~

~~that (X, ^ ) has property K and that (X/'iT*) is regular. Then (X/£)~~

~~i s s ta b le .~~

~~Proof: Let F be closed in (X,^*) and let B£F with c^B)^ F.~~

~~th en 3 x£F-c*(B). By regularity,3 O'^O * 1 such that~~

~~x£o‘<£ c<*(0,)£ 0 llSc*(0,I)£C c#(B). Let~~

~~U=0,x 0 ,,uC c*(0l)x~~

~~C^Cc>(Of) and so o'aC = /. Thus x^ c^(C) and c*(C)^F. By~~

~~Ill.li.S (x/£) is stable.~~

~~III. i;.7 Example. Let X=-£(x,y)| (x,y) = (1,0) or x^O and y> 0^.~~

~~For each (x,y)^(l,0) let£s£ ((x,y)) | £>0^be a base for the~~

~~neighborhood system at (x,y). Let~~

~~£ (x ,y ) j ( x ,y ) = ( 1, 0) o r x > 0 and be a basic neighborhood of 62~~

~~(1,0) for each £ > 0# As is known, (X/^) is Tg but not regular and,~~

~~for any (x,y)^t(l, 0) w ith (x ,y ) £ 0 G.fy', 0-^ C ^ such that~~

~~(x,y)f 0-^^:c(0^)£.0# Let F be closed in (X,^) and let B£F with~~

~~c(B)^F. If 3 (x,y)eF-c(B) with (x,y)=jt(l,0) by the argument used~~

~~in I I I . l j #6 3 such that C£U[B^| implies c(C)^F. Thus~~

~~assume F= c(B)U{(0,l)j• Then 3 U £ ^ ) with U=U-' and~~

~~u[(0,l)]s£c(B ). Let CSU[Bj. If c(C)= F, C£c(B)L>{(0,1)} implies~~

~~(0,1)£ C. But then (0,l)eU[Bj and so u£(0,l)]A B^^ which is a~~

~~contradiction., Thus c(C)^F and by III.I;.5 (X,*^) is stable#~~

~~III# 5e Completely Regular Spaces#~~

~~III. ^.1 Lemma, Let (X/if) be completely regular and let ^iL be~~

~~a uniformity for X such that — ^ * L et Ue. iU ft) and let A€C(X).~~

~~Then 3 V e*UL such that (c(A)xX)/\V£U.~~

~~Proof? Deny the conclusion. Let i t be directed by inverse~~

~~inclusion# Define Si^c(A )*X by S(?) = (ay,xv) where~~

~~L et TTf and 77\. be the projections from~~

~~X*X into X, let S^~ 71) «S and let Sg= 71^ S. Since c(A) is compact~~

~~and S^: 2Z—*c(A), assume that is convergent to Xq. Clearly,~~

~~xQe c(A). That lim S2= x 0 is shown as follows? Let xQ£ 0 6 ^ #~~

~~There is a VeU 5 V[x0~)£=.0 and 3 M eK such that WoWS.V# There exists v’c'U such that v'gW and V* implies S^(V7* ) e W[xQ] . Let w 'sv ’. Then SCw’jrs-Ca^jijX^Oe w' and S-j_(w'eW£xQ]# Then~~

(x0,awi)ew'^W and ) & and so (x^x^Jfi W«*W£.7# Thus w'^V' implies S2(W,)gV£xo] and so lim S 2 = xQ. Since lim S^r; lira S2 = xQ, the net S converges to (x0,xQ). Since U<£%(tr), there is 0 £ ° Y such that (^ ,x 0)eO^O£U. Since lim S = (x^,xQ), 63 there is V £ °IX such that V*SV implies S(Vf)eOO. In particular,

~~S(V) = (a^Xy) <=. OxO^U which is a contradiction.~~

~~I I I . 5.2 Theorem. Let (X/£) be completely regular and let t i be a uniformity for X such that — ^ . Let U and let AeC(X).~~

~~Then H(U)[Aj is a ($)-neighborhood of A.~~

~~P ro o f; By the p reced in g lemma 3 V.ssV"^ ^ such that~~

~~(c(A)xX)/) V£|U/"|U"’/. Since ^ is a uniformity, H(V)[aJ is a~~

~~(id) -neighborhood of A and so it is sufficient to show that~~

~~H(V)[A]S H(U)[A]. Let B£H(V)[A] and let aeA. Then 3 b B such that (b,a)£V. Since V^V"', (a,b) £ (c(A)xX)A V£ Ufi U“‘. Thus~~

~~(b,a)<£ U and so A^U[bJ. Similarly, B£U[A],~~

~~I I I . 5»3 Corollary. Let (X/$ be completely regular, let~~

~~A e.C(X) and l e t Ue'ZlC'k'). Then H(U)£a] is a ^-neighborhood of A.~~

~~Proof? Let be any uniformity for X such that —~~

~~Then ' 2 and so ^ By III.£.3 H(U)[ a] i s a~~

~~^(d)-neighborhood of A. Thus H(U)[ a ] is a -neighborhood of A.~~

~~I I I . 5«h Theorem. Let (X/fcO be completely regular and letbe a uniformity for X with (U) =*£. Then C(X) =C(X) A 4 ”.~~

~~Proof: As above, since %L £= %&)> . Thus~~

~~C(X)n,!c'(^)SC(X)/l4'. Let AeOnc(X) for some 0 £ ^ . There is a~~

~~Ue V.O?) such that H(U)[a]s 6 . By I I I . $ .2 H(U)[a] is a~~

~~^ (20-neighborhood of A. Thus A e C(X)AH(U)|a]£C(X)/)0 and so c(x)Aoec(x)r\ ^(d).~~

I I I . 5 .5 Theorem. Let (X/fc) be stable. Then (X/l) is completely regular if and only if (C(X), C(X) A i s completely » re g u la r. 6h Proofs Assume (X/£) i s com pletely re g u la r and l e t be any uniformity such that ^@ 0 = ^ . By III. 5 .I1 C(X)/a^= C(X)A9t$.).

Thus (C(X), CCx J a ^ ) is completely regular*

~~Assume (C(X), C(X) /T'V) is completely regular. Since (X/V) i s stable, c({x3) is compact for every xeX. Thus, letting i:X-*X by i(x);=-fxj, i[xj^C(X). Since (X/t') is homeomorphic to~~

~~(ifx], i[x]A*?) and (ifx j, i[x] a£)£.(C (X ), C (X )A ^), (x/fc) i s completely regular.~~

~~The next few results are similar in statement and proof to certain results of Caufield [ 3J# in particular, the following theorem: Let (X/U) be a uniform space and let isX-^X by i(x)= ^x|.~~

~~Let A^X. Then c^(i[A] )=^b|^^-BSc({'x}) for some x£ c(A)} .~~

~~The first lemma is due to Davis £ 6j .~~

~~III. £.6 Lemma. Let (X/fc) be regular, let U <£. and let x<£X. Then 3 V such that VoV[x) ^ U[x].~~

~~P ro o f: There i s a 0£*¥ with xe 0£SU£xJ. By regularity~~

~~V = 02x 02u [0-.c(01) ] x [o - c(01) ] u ^ c(02 ) ^ (! c(02). Clearly, 7e~~

~~V £x] = 02 and VoV[xJ £ O S U fxj.~~

~~III. £.7 Lemma. Let (X/£) be regular and let i:X— by i(x) = <£xj. Let A£X. Then k(i£A]]) =^b|^B ^.c(-/x]) for some xe c(A)}.~~

~~Proof: Let Bek(ifA^j). Since is open, k(i[A]))^ ^ and so~~

~~B ^ . Let x<=B and let U €^(^). There exists H(U)[ b]A i[A j.~~

~~Then x £ Bcu[a] and so x £/){vyi} | U 6 c(A). Now let U £ % C^) and l e t V = V” ‘ 6 tiff?) with V«V£xj£E [x]»U There exists~~

~~£ a H(V)[b]f| i[A.l» Then x£B £y[a'j and so a’fiVCxJ. Thus BGLV[a)^VoV[x]<^U£xJ and so B^A {u£x]| U e.% (*&}=*c({x j).~~

Now let j^fBSc ({x}) fo r some xec(A). Let Ue % (X) with U=U-# and U open. It is sufficient to show that H(U)[bJ/"| Since x£c(A), there exists acA such that xeU[a], Since (X/c) is stable and U£a] is open, c({x3)$l U£a3. Thus B£.U[a3 and, since a £ U[Bj. Thus {&l G H( U) [b] A i W •

~~I I I , £.8 Theorem* Let (X,*^) be completely regular and let~~

~~' iX~ >X by i(x)=:£x|e Let A£XS Then c(itA ])=jB|^ 7t B £ c ( |x ^ ) f o r some x e c (A )}.~~

~~Proof; By HIo5«7j> ^ b) ^^B^c(-[x3) for some xec(A)l; = k(i[A[J c(ifAj ) . L et %i be any uniformity for X such that ^(£^)=‘^ ®~~

Then and so Let be the closure in^(2£)*

~~Since c(i(Xj)£=<^(i[Aj) and so, by Caufield1s theorem mentioned above III.f>. 6, cCi/Aj )S.^b]^^B£c(-[x}) for some x ec(A)}.~~

~~III. 5.9 Theorem* Let (X/b) be completely regular and let i:X->X by i(x)=-{x^. Let A be generalized closed in X. Then i[A^ is~~

~~A A generalized closed in (X,x).~~

~~Proof; Let i[A]£0 with O c ^ , Let 0 = 1“'0 3 . Then A~~

~~^B£c({x3)o Since x£i-1 0 ] , 3 U=-U" £ ‘U (*V) w ith H(U)[j; x;[|<= 0 .~~

~~Then B£ic£x3 ^U[xJ and, since U=U_I and B=£^, xeU0B]. Thus c ( i W ) S 0 .~~

~~I I I . 6. Compact Spaces. A A III. 6.1 Theorem. Let (X/£) be a topological space with (X,cO compact. Then (X/iT*) i s compact.~~

~~P ro o f; L et £ 0%. i 44 e ^ be .a 't*-open cover of X. Then, for 66 each j^A^X, for some «6A . Thus X< Ljl £=U^(C&)~ \<*-£ and so, since t 3 ^ a <= A such th a t X --^]^ U((Q^)“.~~

~~Clearly, X S \Jf 0*^.~~

~~III. 6.2 Example. Let X be infinite and let x0£ X. Let~~

~~^=^o]o = ^ or xQ eo}. Clearly, (V) consists of all supersets of~~

~~B = U {{x,x0]x |x £xj. Let 6 with 0?^ and Jand l e t~~

A£0 with A^£^. ThenH(B)[ a [|<=.6. By the definition of B it is clear that H(B)[{x^j] = X -$? so that {xQ^ £H(B)[ a }s 0. Then H(B) 6 and so 0 — X-^j or 0 = X. Thus C |/L x^and (X/V) is compact while (X,V) is not compact.

~~The remainder of this section uses universal nets. A net SsD—>X is universal provided that S is eventually in A or C A f o r ev ery AS X.~~

~~A topological space (X/f) is compact if and only if every universal net converges. (See Kelley [lo].)~~

~~III. 6.3 Definition. Let (X,V) be a topological space and.let~~

~~S:D-*X be a net. Then lim S= £x|x£0 0/\S(d)^j^ eventually^ and lim S = £x [ x £ 0 C ^^O /lSC d)^^ frequently^-.~~

~~III. 6.1; Lemma. Let (X,V) be a topological space and let~~

~~S:D->X be a universal net. Then lim Stslim S.~~

~~Proof: Clearly, lim Sslira S. Let xelim S and let x£0.~~

~~Since S is universal, S is eventually in 0" or S is eventually in~~

~~C(0~). Then, since S is frequently in 0~, S is eventually in 0~".~~

~~Thus x £ lim S.~~

~~III. 6.5 Lemma. Let (X/V) be a compact topological space and let S:D-*X be a net. Let Ue^(V). Then 3 such that d ?dQ implies lim SSU[s(d)J. 67~~

~~Proofi Since X is compact, there exist O p.-.jC^d^such that~~

~~X£UO. and Q O^O^U. Let N={i[lim For each i € N 3 di gD such that d^d^ implies S(d)AO^ There is a dQe D~~

~~such that d0^ d^ for each ieN . Let d^d0 and let xelim S. There is an 0^x£0^, Then ie N and S(d)/\ 0 ^ ^ . Thus xt 0^£U[s(d)] and so lim S£U[s(d)][.~~

~~III. 6.6 Lemma. Let (X,^ be a compact topological space and l e t S;D—& be a net. Let Vetlift)* Then 3 dQ£D such that d'^d0 im p lies S (d )£ U^lim s][.~~

~~Proof? Assume without loss of generality that U is open in~~

'Y x 'Y , For each x ^ U[lim 3 xe. 0^.4^ and d^e. D such that d?dx implies S(d)/^ Since (U^lim sj) is closed in X and so compact, there exist x^,...jXj^ such that ^ (U lim S )*~U 0X^. There exists a dQ£D such that d ^ d ^ for each i. Let d^d0. Since

~~S(d)f| 0Xi= ^ for each i, S(d)<= £( O o^J^ujlim s ] .~~

~~III. 6.7 Theorem. Let (X/T) be a compact topological space. A A . Then (X/r) is compact.~~

~~Proof? Let S?D-*X be a universal net and let A= lim S =lim S.~~

~~Let Ae 0 6 ^ . Then 3 U £.%.&') such that H(U)[a]^0. Using III.6 .$ and III.6.6 3 dQe D 3 d^dQ implies S(d) 6 H(U)[a], Thus lim S=A.~~

~~III. 6.8 Theorem.- Let (Xft) be a stable topological space.~~

~~Then (X/£) is compact if and only if (X/£r) is compact.~~

~~Proof? This follows immediately from III.6.1 and III.6.7o~~

~~III. 7. Locally Compact Spaces.~~

~~III. 7.1 Theorem. Let (X/£) be stable. Then C(X) is open in~~

~~(X/£) if and only if every x£X has a closed compact neighborhood. 68~~

~~P roof? Assume C(X) i s open and l e t x e X . Then £ x je C(X) and so~~

~~3 U £ lift) such that H(U)[|x]]^C(X). Thus Ufx]cC(X) and so c(U[x]) is a closed compact neighborhood of x.~~

~~Now assume that each point has a closed compact neighborhood.~~

~~Let A£ C(X). For each xec(A) 3 a closed compact neighborhood of x,~~

N . Since c(A) is compact, there exist x-^ •• .jX^e c(A) such that n c(A)£U(Int Nx^. Thus 3 0 and F closed and compact such that c(A)£0£F. Let U = 0*0 1>£c(A )* £ c(A). I f B £ H(U)[ a ], th en

~~B£U[A]£0. Then c(B)£F and so c(B) is compact. Thus H(U)[A]£C(X).~~

~~E l. 7o2 Corollary. Let (X/^) be a regular or Tg topological space. Then (X,£) is locally compact if and only if C(X) is open in~~

~~( x A .~~

~~Proof: In a regular or Tg space, the closure of a compact set is always compact. Thus 111.7*2 follows immediately from the preceding theorem.~~

~~H I. 7*3 Theorem. Let (X/£) be completely regular. Then (X/t*) is locally compact if and only if each Ae C(X) has a closed compact~~

~~‘T -neighborhood.~~

~~Proof: (Necessity) Let A£ C(X). Just as in 111.7*1 3 closed compact F and 0 such that c(A)^0£F. By H I.1.11~~

~~(F,F/TV)£(X/£) and by 111.6.7 (F,F/V£ ) is compact. Let~~

~~U = 0x0O^c(A)xCc(A). As above H(U)[ a] ^ F += F. By I I I . 2 .2 F"**is A ^ closed in X and by III.fj.3 H(U)[ a 3 is a “^-neighborhood of A.~~

~~(Sufficiency) Let i:X—»X by i(x) = £x^. Since (X/f) is stable, i t i s s u f f ic ie n t to show th a t (i[xj,i[xj f\*$) is locally compact.~~

Let |x ^£ i[x j and let I be a closed compact ^-neighborhood of £x^. 69 By IIIo£.9 i[x] is generalized closed in X. Since N is closed, irxiAft is also generalized closed in X. Since i[x]AN i s a generalized closed subset of a compact set, i[xJ/\N is also compact.

~~Thus i [xj f\ N i s a compact i{x] (\°X -neighborhood of {x}»~~

~~III. 7®U Theorem. Let (X/fc*) be completely regular. Then (X,'V) is locally compact if and only if (C(X),C(X)A'£) is locally compact.~~

Proof: (Sufficiency) Since (X/b) is stable, letting i:X-^X by i(x)=sfxj, (X,-Qr)cv(i[x],i[x] /)^)€(C(X),C(X)/1^). By III.5.9 i[x ] is generalized closed in X and so i[x3 is generalized closed in C(X).

~~Thus (X,“V) is locally compact.~~

~~(Necessity) Let A£C(X). By III.7.3 3 a closed compact~~

~~‘’T-neighborhood of A, N, and so 3 U &%&) such that H(U)[a]SN. By~~

~~1 1 1 .7 .2 , C(X) i s open and so 3 V e with V^U such that~~

~~H(V)[A]SC(X). Let %i be any uniformity such that tf’t,(ii)=-‘5c'. By~~

~~I I I . 2 3 such that Ac8sH(V)[A]. Since (X,^$)) is regular, there exists an O -^^^.) such that Ae c^(&j_)£ 0. Then, sin c e ri ( $ L ) ^ ‘r^,0-^£‘'V and c(0^)^c^(8^). Moreover,~~

~~0 ( ^ ) 2 H(V)£ a ] s C(X) a N. Thus c ^ ) is a compact~~

~~C(X) A '^-neighborhood o f A. CHAPTER IV~~

~~PSEUDO-METRIZABILITY AND THE HYPERSPACE OF A TOPOLOGICAL SPACE~~

~~IV. 1. The Hausdorff Pseudo-metric.~~

~~This section describes the version of the Hausdorff metric appropriate when the hyperspace of a set is the set of all subsets.~~

~~In published work this material usually seems to be passed over completely or mentioned vaguely.~~

~~IV. 1.1 Definition. Let (X,d) be a pseudo-metric space with d-1. ForA,B£'X,~~

~~1 if A = ^ and B ^ / or A^jrf and B=^~~

i n f k £ \ (A,B)e H(VC)^ otherwise A • The function d w ill be called the Hausdorff pseudo-metric generated by (X,d). Note that, since d-^1, ^ £ j (A,B)£ H(Ve ) ^ ^ provided A and B are both empty or both non-empty. In addition, A since 1, clearly d£ 1.

~~IV. 1.2 Theorem. Let (X,d) be a pseudo-metric space with d ^ l.~~

~~Then d is a pseudo-metric for &•~~

Proof; Clearly, for everyA,B£X, d(A,B)^-0, d(A,A)=0 and d(A,B) = d(B,A). Let A,B,C£X. If any two of A,B,C are equal, it is clear that d(A,B)^ d(A,C)+3(C,B). Thus assume A,B, and C =>.re a ll d i s t i n c t .

Case I : One of A,B,C is Then d(A,C) = l or d(C,B)=l and so d(A,B)< l^d(A,C)+cl(C,B). 70 71 Case II: None of A,B,C is ;zf. Let d(A,C) = r^, let d(C,B)=rg and let £ > 0. Then (A,C)£ H(Vr^+<^) and (C,B) £ H(Vr^ ^ ) . Thus

~~B)£H(Tri+% ).H(Tr2^)£H (T ri+ ^ V r2+%)SH(Vrrr2t(.) and so d(AaB)< rj+Tg-f £ • Hence, d(A,B)^ ci(A,C)+ci(C,B).~~

The next result is the motivation for the standard definition of the hyperspace of a semi-uniform space. This result, by constructing a specific hyper-pseudo-metric, contains, im plicitly the fact that the hyperspace of a pseudo-metrizable uniform space is pseudo-metrizable.

~~IV. 1.3 Theorem. Let (X,d) be a pseudo-metric space with d^l.~~

~~Then U4 -V < £ .~~

~~Proof; Let 6 > 0. Clearly H(Vd,^)£V £ e and~~

~~S. H(V^ £ ). Since such sets are basic,. 2^ •~~

~~IV. 1.1* Theorem. Let (X,‘V) be a stable topological space.~~

~~Then (X,^) is pseudo-metrizable if and only if (C(X),C(X)/\^ ) is pseudo-metrizable.~~

~~Proof? (Sufficiency) Since ( l/t) is stable, (X/£) is~~

~~A homeomorphic to a subspace of (C(X),C(X) A*Y).~~

~~(Necessity) Let d be a pseudo-metric for X with and d £ l. By Ill.^oh and IV.1.3 C(X)A~~

~~(C(X),C(X)/1 ^ ) is pseudo-metrizable.~~

~~IV. 1.5 Theorem. Let (XyV) be a stable topological space and let iiX-^X by i(x) = |x^. Then (X/£) is metrizable if and only if~~

~~(C(X),C(X) a 4 t ) is pseudo-metrizable and i[x} is closed in C(X).~~

Proof: (Necessity) By IV. 1.1* (C(X),C(X) f \ $’) is pseudo- metrizable and by III. 1^.2 i[X3 Is closed in X. Since i£x]£C(X), i[x][ is closed in C(X). (Sufficiency) By IV.1.1* (X,‘V) is pseudo-metrizable. Thus

~~III.5.8 applies and c(i[xJ)-=^B|p'for some x£X^. Clearly, c(i[x])£C(X) and so i [x]=C(X)D c(ifx])= c(i[x]). By III.1^.2 (X/£) is~~

~~IV. 2. Pseudo-Metrizability Of Uniform Topologies.~~

~~IV. 2.1 Lemma. Let (X,&) be a uniform space and let~~

~~i £ Then 3 a pseudo-metrizable uniformity with~~

~~{ U j i c P~~

~~Proof: For each i 3 a pseudo-metric d^ in the gauge of with~~

~~. Let 1 C - V { ^ ( i £ ?]• Then *U* is a pseudo- metrizable uniformity with i e P^ ~ cU^‘^ lU..~~

~~IV. 2.2 Definition. Let (Xyfy) be a uniform space. (X,&) is uniformly first axiom if and only if 3 { U^| i €. P^G. % such that~~

~~[u^[x])iep] is a neighborhood base at x for each xeX.~~

~~IV. 2.3 Theorem. Let (X,2l) be a fine uniform space. Then~~

~~(X,^(fc)) is pseudo-metrizable if and only if (X/U.) is uniformly first axiom.~~

~~P ro o f: Assume (X,is pseudo-metrizable and let d be a pseudo-metric with ’%i = £V(‘2^). Then, since (X,20 is fine,~~

~~■[vd _l J n e p |^ ^ . C le a rly £vd,-{, C x 3 )n£ *,s a neighborhood base at x for every xeX.~~

Assume th a t (X,%) i s u niform ly f i r s t axiom. L e t^ U ^ i e P ^ be a family such that ^Uj_£x3| i£ P^ is a neighborhood base at x for each xeX. By IV.2.1 3 a pseudo-metrizable uniformity slC '<=-°iL w ith

~~{ u j i e 7 ? $ G . 0l t . Since ^ (#*) £ ^(40. Let x£ 0c .~~

Then 3 ieP such that U^lXl^O. Thus 0 (U*) and (It*)— ^(U -). IV . 2.h Theorem. Let (XfU.) be a uniform space. Then (X ,^^)) is pseudo-metrizable if and only if there is a uniformity V,’* such th a t f **£(U) — and (X,W.tf) is uniformly first axiom.

~~Proof; If (X,^#.)) is pseudo-metrizable, let W* be the fine uniformity for T(^)« Then ^(*20 = , ffU .^ eU* and (X,^) is uniformly first axiom.~~

Now assume 3 a uniformity such, that 3 ^ — and (X$L*) is uniformly first axiom. Let V be the fine uniformity f o r ‘Y ($k). Since 1k*^f\/ , it is clear that (X/V) is uniformly first axiom. Thus °X (V) = ^ ('20 is pseudo-metrizable.

~~IV. 2.5 Corollary. Let (Xft) be a topological space. Then~~

~~(X/^) is pseudo-metrizable if and only if 3 a uniformity %L f o r X such that and (X,^) is uniformly first axiom.~~

~~P roof; This i s immediate from IV .2.1*.~~

~~IV. 3* Pseudo-Metrizability Of Fine Uniformities.~~

~~IV. 3.1 Lemma. Let (X,v2t) be a uniform space; let A^X with~~

~~(A,A*A DU) discrete. Then 3 V=V~*e^ such that VCajAV£a1J — ft f o r ev ery a?fc a* in A.~~

~~Proof; There exists U eW such that A*Af\U ^A ^and~~

~~3 V=V-,£^U such that V»VSU. Let a,a*e A with a^ a' and Suppose that xe V[a}A Vfa1] • Then (a,x)eV and (x,a')eV and so~~

~~(a,a')e VoV^U, a contradiction.~~

~~IV. 3*2 Lemma. Let (Xftt) be a uniform space; let A£X with~~

~~(A, Ax A /\cii) discrete. Then c(A)=(^^c(-fa^)|aA A^.~~

~~Proof: By the preceding lemma 3 V — V^e'W. such that~~

~~V[a] A V[a^J =, for every a^a* in A. Then it is clear that 7k |v£x]flA |^l for every x £ X. Thus ja £ i s a locally finite~~

~~family of sets and so c(A) = U ^c({aj)|a £A^.~~

~~IV. 3*3 Lemma. Let (X,%) be a stable topological space and let~~

~~D= U<£c({x} )J c({x?) Then D ~{x]c({x3)£^J»~~

~~Proof; If c({x3) then x£ c(-{x3)^D. Let xaD. Then~~

3 y such that x£c(fy?) and c( {$)&*

~~xec({y}), c({x3)= c(^y?). Thus c ( { x ] ) ^ ,~~

~~In the remainder of this section the following fact is needed;~~

~~Let (X,5# be pseudo-metrizable. Then is the fine uniformity f o r and (X,U(^)) is complete. (See Kelley [loj.) The metric space version of the next three theorems (in~~

~~particular, IV.3.6) appears in Atsuji £l} and Rainwater [ 15J . The~~

~~following proof was discovered independently.~~

~~IV . 3.I1 Theorem. Let (X/£) be a pseudo-metrizable topological~~

~~space; let D — Ufc({xJ)j c({x?)£ \ and l e t F = CD. Assume th a t~~

~~(X, ?*(«*)) is pseudo-metrizable. Then (F ,F /^ ) is compact.~~

~~Proof; Let d be a pseudo-metric such that Since F~~

~~is closed and (X, %(?£)) is complete, (F,F*FA Cli^c)) is complete. Thus it is sufficient to show that (F,FxF/A%C£)) is totally bounded.~~

Deny. Then 3 A£F such that J A) — H0 and (A, Ax A A'M-CW) i s d is c r e te . Assume A = {a^)i£ P ^ and i =£j im p lies a -j^ a ^ . By TV.3 .1 3 V —Y~i e. %(.&) such that V[a^ ft V£a£ = $ if i^t j. For each

~~i 3 0 such that Sr (a^cvfa.j'] • Also, since c({a 3 Sj_(ai)-c({a^ ). Pick o< ^ such that ki~~

~~S^=- SJCi(a;jL)«SjCi(ai )o. Let Sg = Cc(A)xCc(A)o Since~~

~~c(A )= jjt c ({a^f ) , W— S^L* S 2 G.2i o Thus 3 n such that V^SW. 75~~

~~Then (anjyn)eW, Clearly (sn jJeS g since ane c(A). Hence~~

~~(an>yn)e ski(ai)*ski(ai) £or sorae is Since~~

~~Sk i(ai)c=V£ai], S^aJSV O j and V[a£JAV[anl = i^ if i=*n, i= n .~~

~~Then yne Sk (an) and so d(an,yn)< kn. This contradicts the choice XI o f kn .~~

~~IV. 3*5 Theorem, let (X/V) be a pseudo-metrizable topological spacej let B — u[c({x^)j c(-fx}) £.*¥ | and l e t F = £ D . Assume th a t~~

~~(F,F is compact. Then (X,'UC^)) is pseudo-metrizable.~~

~~Proof: Let d be a pseudo-metric with ‘T o Let~~

~~B=u{c(-fxJ)xc(.{x3)| x £ d ]. By IV .3 .3 B C le a rly BSU f o r every UetiCtO. For each i let C— U^Sj^xJxSi^x) | xe F^ and let~~

~~V±-B UC±. C le a rly { v j i e P $£ ft ftr) and so, by IV. 2 .1 , 3 a pseudo-metrizable uniformity %LK£=.%($') w ith £v-jJ i t L et~~

~~UcUOf). For each xeF 3 £x> 0 such that Se^x)*Sej<(x)^U.~~

~~Since F is compact, 3 a finite set ISF such that~~

~~F £ U{Sej,(x)j x el^. Pick n such that k <-| min-[£xj x e l| and let (x,y)eVn. If (x,y)eB, then (x,y)eU.~~

~~Suppose that (x,y)aCn. Then 3 zeF such that (x,y) e Sj_(z)xSi,(z). in There is a we I such that zeSe^w). Then d(x,w)^ d(x,z)+d(z,w)< — £-u< and, similarly, d(y,w)< £w .~~

~~Then (x,y)e S£w(w)xS£w(w)s.U. Thus Vn^U and CU.(X)£'U*«~~

~~IV. 3«6 Corollary. Let (X/i?) be a pseudo-metrizable topologi cal space, let D = U|c('[x})\c({x}) €.6Y*| and let F = £D. Then~~

~~(X, is pseudo-metrizable if and only if (F,F/"|^) is compact.~~

~~Proof: This summarizes IV.3«li and IV.3*5»~~

~~IV. 3*7 Theorem. Let (X,^) be a fine uniform spacej let D = u|c({x^)j c(£x3) €. and l e t F =~~

~~(Fj FA^YC^)) i s compact® Proof; (Necessity) By IV.2.3 ( XJil) is uniformly first axiom.~~

Also (U)) and, by IV.3.6, (F,F A^C2l)) is compact*

~~(Sufficiency) By IV®2®3 (X, is pseudo-metrisable® Then~~

~~‘U. = sli(^ (iii)) and, by IV.3.6, (X/U_) is pseudo-metrizable®~~

~~IV* h» M etrizability Of The Hyperspace®~~

~~IV* hoi Theorem. Let (X/V) be pseudo-metrizable; let;~~

~~D = (yjc({x3)|c(-[x3) and let F = (2D® Assume that (X/£) is first axiom. Then (F,FfrY) is compact®~~

~~Proof; Let d be .a pseudo-metric such that . Since~~

~~(X, '2i(ft)) is complete and F is closed, it is sufficient to show that~~

~~(F,FxF/^fy(°fcr)) is totally bounded. Deny. Then 3 A<~F such that~~

~~JA |= and (A,A*A A V.fft')) is discrete® Assume that A={a^|i£P3 and i^ j implies a ^ a ^ . By IV.3.1 3 V = V “ *e % C’fc') such that i implies V[a^ f\VLa.p = Since (X/f) is first axiom,~~

~~3 £ u .jjie Psuch that U^cV for every i and^H(U^)[a]]} ie P^ is a neighborhood base for at A. Let ieP . Since c({a $)£*)?t~~

~~U£af] -c(âj3 )^ô Thus 3 0 such th a t Uâ^l - S j^ a ^ ) ^ :^ and s £t(ai)*s$t(ai)—Ve Let S-^sU'^S^ (ai )xS^(ai)|i e pj. Let~~

~~S2 —Cc(A)x/^c(A). Since c(A)= c({a^5 ), W sS^Sg £ Ufa*). Then~~

~~3 neP such that H(Un)[A]£ H(W)[Aj. In particular, Un[A]<^VJ[kj.~~

~~Let x £Un[anl-S^(an). Then 3 j such that (a^,x)eW. Since a^ eA~~

~~x e S^-, (aj) — V[anlAVCajl. and so j=n. But then xe Sg (an)~~

~~0 which is a contradiction.~~

~~IV. i|.2 Theorem. Let (X^) be a pseudo-metrizable topological~~

~~spacej le t D = U^c(*rx ]) j c(£>c5) £ ^ j ^ a n d l e t F«=-CD. Then the following are equivalent:~~

~~a ) (X, %&)) is pseudo-metrizable~~

~~A A b) (X, %(&)) is pseudo-metrizable~~

~~c) (F ,F /^) is compact~~

~~d) (X/&) is pseudo-metrizable~~

~~Proof: The equivalence of a, b and c has been established in~~

~~IV. 1.3 and IV.3.6. Clearly b=^d. By IV.li.l d=^c.~~

IV. 1|.3 Corollary. Let (X/V) be a stable topological spacej let D= U^c(-{x5)| c(^x3.)£^^ and let F= CD. Then (X/c) is pseudo- metrizable if and only if (X,T) is pseudo-metrizable and (F,F fY)^) is compact.

~~Proof: If is pseudo-metrizable, then, since (X/£) is essentially a subspace of (x/t), (Xs"t) is pseudo-metrizable® Thus~~

~~IV.2 applies and (F,FAci') is compact.~~

~~I f (XfZ) is pseudo-metrizable and (F,F is compact, by~~

~~IV.lj.2 (X/l) is pseudo-metrizable.~~

~~IV. U.l*. Theorem. L et (X,*l 0 be a fine uniform space. Then (X,2^) is pseudo-metrizable if and only if (X^fy)) is pseudo-metrizable.~~

~~Proof: (Necessity) This is clear since (X,1^) is pseudo- y\ A m etriza b le i f and only i f (X/li.) i s p seudo-m etri 2a b le .~~

~~(Sufficiency) Since (X ,^ (lU.)) is homeomorphic to a subspace of (X, ^(ft)), (x>‘T(2t)) is pseudo-metrizable. Thus H- ) and so by IV.ii.2 (X,^) is pseudo-metrizable. 78~~

~~IV. Corollary. Let (X,^) be a fine uniform space. Then~~

~~(X/U.) is pseudo-metrizable if and only if (X,‘£'(‘20) is pseudo- metrizable.~~

~~P ro o f: I f (X/UL) is pseudo-metrizable, then iU—'U(o£fy0)<> Thus~~

~~and, by IV.U.U, (X, ”£ ($ 1 )) is pseudo-metrizable.~~

~~I f (X, W) is pseudo-metrizable, by IV. 1*.2 (X, is pseudo-metrizable. Since (X/li) i s f in e , %L~(SU.(s)/tyl)) and so (X/tt) is pseudo-metrizable. CHAPTER Y~~

~~SOME QUESTIONS~~

~~The following list contains a number' of open problems* The first six are general topics for study; the last ten are fairly specific questions or conjectures.~~

~~1) Study multivalued functions in the context of semi-uniform spaces or topological spaces.~~

~~2) Develop a theory of "proximity-like" relations induced by semi-uniformities, semi-uniformities with property K and HK semi-uniformities.~~

~~3) Define and study semi-metrics, semi-metries with property K and HK semi-metrics.~~

~~it) Study topological properties of the hyperspace of a topolog ical space. (Many properties such as hyper-normality or hyper regularity were not even mentioned in this paper.).~~

~~£) Given a topological space (X,4^), let H be the cannonical semi-uniformity constructed in I.1.23* Study (X,4^^ )) as the hyper space of (Xff).~~

6 ) Study the role of completeness and total boundedness for semi-uniform spaces with property K and HK spaces. (It might be useful or necessary to modify the definitions: perhaps, for example, a semi-uniform space (XJU.). is totally bounded if and only if every filter has a Cauchy superfilter.)

~~79 . 80~~

~~7) Given an HK space (X,%), is (X,^(^.)) compact if and only if~~

~~(X,%) is complete and totally bounded? (It is not hard to show that completeness and to tal boundedness are necessary.).~~

~~8) Given (X/£) compact and regular, does there exist a unique~~

~~HK semi-uniformity which generates ? (Using a lemma similar to~~

~~III.5.1 it is possible to show that there is a unique HK semi uniformity which generates^ and is contained in %&).)~~

~~9) Is every entourage in an HK semi-uniformity a neighborhood of the diagonal?~~

~~A A, 10) Give an example of an HK space (XJli) for which (X,*ii.) is not an HK space.~~

~~11) Give an internal characterization of HK spaces.~~

~~12) Are the two conditions which characterize HK spaces in~~

~~II.2.12 independent? (Note that, if X is infinite and ^ is co- finite, for every A£ X and U(t0 A(U) i s a “^-n e ig h b o rh o o d o f A but (X,^(T)) is not an HK space.)~~

~~13) Give an internal characterization of HK topological spaces.~~

~~(An attractive conjecture: (X, is an HK space if and only if~~

~~(X,^) is normal.)~~

~~1U) Give an example of a topological space (X/£) such that~~

~~(X,ti (^)) is an HK space but not a uniformity. (For a characteri zation of spaces for which (X,is a uniformity and for some strong normality properties which may be relevant see Mansfield [l2],~~

~~Cohen [51 and Bing [2^.)~~

~~1J?) Is a stable topological space (X/i") regular if and only if~~

~~(C(X), C(X)fi't) is regular? 16) Given a topological space (X,^), is (X,^) compact if and «n!Ly i f ( X, '£*) is compact? BIBLIOGRAPHY~~

~~1. M. Atsuji, Uniform Continuity of Continuous Functions of Metric Spaces, Pacific J. Math. vol. 8 ( 195?8), pp.. 11-16.~~

~~2. R. H. Bing, Metrization of Topological Spaces, Canadian J. Math. vol. 3Tl9'£0, pp." 17>186.'~~

~~3. P. J. Caufield, Multi-Valued Functions and Uniform Spaces, Thesis, The Ohio State University^ Columbus, Ohio, 1967.~~

~~ii. E. Cech, Topological Spaces, Inter science, 1966 ®~~

~~$» H. J. Cohen, Sur un Probleme de M. Dieudonne, C. R. Acad. Sci. Paris vol. 231i (1952), pp. 290-292.~~

6 .- A. S. Davis, Indexed Systems of Neighborhoods for General Topological Spaces, Amer. Math. Monthly voi. 68 (1961), pp .1 686-893*

~~7. N. A. Friedman and R. C. Metzler, On Stable Topologies, Amer. Math. Monthly vol. 76 (1968), pp. ii93-U9o•~~

~~8 . C. J. Himmelberg, Quotient Uniformities,, Proc. Amer. Math. Soc. vol. 17 (1966), pp. 1385-1388.~~

~~9. J. R. Isbell, Uniform Spaces, Providence, R. I., 196U.~~

10. J. L. Kelley, General Topology, Van Uostrand, 19!?5>*

~~11.- N. Levine, Generalized Closed Sets in Topology, to appear.~~

~~12. M. J. Mansfield, Some Generalizations of Full Normality, Trans. Amer. Math. Soc. vol. 86 (1957 J, ~PPi;89-5>0j?,~~

~~13. E. Michael, Topologies on Spaces of Subsets, Trans. Amer. Math. Soc. vol. 71""(lp^l'T, pp. 1 5 2 - 1 6 2 .~~

~~111. S. A. Naimpolly, Separation Axioms in Quasi-Uniform Spaces, Amer. Math. Monthly vol. 7l| (l967), pp- 253-23141~~

~~16. J. Rainwater, Spaces Whose Finest Uniformity is Metric, Pacific J. Math. vol. 9 (1959), PP^’~~