Maximal Pseudocompact Spaces

Comment.Math.Univ.Carolinae 35,1 1994127{145 127

Maximal pseudo compact spaces

Jack R. Porter, R.M. Stephenson Jr., R. Grant Woods

Abstract. Maximal pseudo compact spaces i.e. pseudo compact spaces p ossessing no

strictly stronger pseudo compact top ology are characterized. It is shown that sub-

maximal pseudo compact spaces whose pseudo compact subspaces are closed need not

b e maximal pseudo compact. Various techniques for constructing maximal pseudo com-

pact spaces are describ ed. Maximal pseudo compactness is compared to maximal feeble

compactness.

Keywords: maximal pseudo compact, maximal feebly compact, submaximal top ology

Classi cation : Primary 54A10; Secondary 54C30

x1Intro duction

A top ological space X is called pseudocompact if every continuous real-valued

function with domain X is b ounded; it is called feebly compact , or, lightly com-

pactifevery lo cally nite collection of op en sets is nite. Both classes of spaces

have b een extensively studied; see [GJ] or [PW], for example. The following

well-known relationships link these classes of spaces:

1.1 Theorem. A completely regular Hausdor space is feebly compact if and

only if it is pseudo compact.

1.2 Theorem. Every feebly compact space is pseudo compact, but there are

pseudo compact spaces that are not feebly compact.

A pro of of 1.1 and the rst part of 1.2 can b e found in 1.11 d of [PW]; examples

witnessing the second part of 1.2 can b e found in [St ], and one of these app ears

as problem 1U of [PW].

Let and be two top ologies on a set X . If we say that is an

expansion of and that is a compression of . If n 6= ; then is a proper

expansion of and is a proper compression of . If A X , the closure of A in

X; will b e denoted bycl A. If only one top ology on X is under discussion,

we write cl A or cl A instead of cl A. Similar conventions apply to closures in

subspaces.

Let P be a top ological prop erty. A space X; is said to be a maximal P -

space if X; has P and if is a prop er expansion of , then X; do es not

have P .

The research of the third-named author was supp orted by Grant No. A7592 from the

Natural Sciences and Engineering Research Council of Canada.

128 J.R. Porter, R.M. Stephenson Jr., R.G. Woods

Maximal P -spaces have already received considerable study; see [R] and the

ve pap ers by Douglas Cameron listed in the references. Related to the ab oveis

the notion of a strongly P -space; a space X; with P is said to b e strongly P

if there is an expansion of for whichX; is a maximal P -space see [Ca ].

In this pap er we study the class of maximal pseudo compact spaces and consid-

erably extend the previously known results ab out this class. In x2wecharacterize

maximal pseudo compact spaces, and provide counterexamples to other plausible

candidates for characterizations. In x3we pro duce examples of maximal pseudo-

compact spaces and strongly pseudo compact spaces, and describ e ways in which

maximal pseudo compact spaces can b e constructed. In x4we study the relation

between maximal pseudo compactness and maximal feeble compactness. We nish

the pap er by p osing some op en questions.

In the remainder of this intro duction we present some known concepts and

results. We will make considerable use of results app earing in [PSW ] and [PSW ],

1 2

which are \companion pap ers" to this.

1.3 De nition. A top ological space is submaximal if its dense subsets are op en.

We collect a miscellanyof facts ab out submaximal spaces b elow; 1.4 a and

1.4 b are resp ectively Theorems 15 and 14 of [R].

1.4 Theorem.

a A maximal pseudo compact space is submaximal.

b A maximal feebly compact space is submaximal.

c If S is a dense subspace of the submaximal space X , then X n S is a closed

discrete subspace of X .

If D denotes the set of dense subsets of a space X; , then bya dense ultra lter

on X; we mean an ultra lter on the p oset D ; . By Zorn's lemma any

nonempty subfamily of D that is closed under nite intersection is contained in

some dense ultra lter on X; . If S is a collection of subsets of X; , then

S denotes the top ology generated on X by [S, i.e. the smallest top ology

containing b oth and S . If S = fAg resp. fA; B g, we write A resp. A; B

instead of fAg resp. fA; B g. If S is a subset of X , then jS as usual

denotes the restriction of to S . Observe that jS = S jS and jX n S =

S jX n S .

If X; is a space then R resp. RO will denote the set of regular

closed subsets resp. regular op en subsets of X . The semiregularization s of

the top ology is the top ology on X for whichRO is an op en base. Observe

that R s =R and ROs =RO . See Chapter 2 of [PW] for pro ofs of

these claims and a detailed discussion of semiregularizations.

The following is 2.23 of [PSW ] which in turn is drawn from exercises 20 and

22, pages 138 and 139 of [Bo] and immediate consequences thereof.

1.5 Prop osition. Let V b e a dense ultra lter on X; . Then:

a X; V is submaximal, and its dense subsets are precisely the memb ers

of V .

Maximal pseudo compact spaces 129

b RO =RO V and s V = s .

c X; is feebly compact if and only if X; V is feebly compact.

d If X; is T and p 2 X then fpg2 if and only if fpg2 V

e If is a submaximal expansion of and if = s , then there is a dense

ultra lter V on X; such that = V .

In 2.2 of [PSW ] maximal feebly compact spaces are characterized as follows.

1.6 Theorem. A space X; is maximal feebly compact if and only if it is feebly

compact, submaximal, and all its feebly compact subspaces are closed.

If X; is a space then C X; will denote the set of real-valued continuous

functions on X; ; we write C X if it is unnecessary to sp ecify . The set of

b ounded memb ers of C X; is denoted C X; orC X . The weak top ology

induced on X by C X; is denoted by w ;thus w is the top ology on X for which

coz X; is a base. [Here coz X; denotes the set of cozero sets of X; , i.e.

the set fX n f 0 : f 2 C X; g. If no ambiguity can arise ab out the top ology ,

we write coz X instead of coz X; .] We call X; completely regular if = w .

Completely regular spaces need not b e T ; observe that a space is Tychono if and

only if it is completely regular and T , but X; w need not b e T even if X;

1 1

is. However, the pro of of 1.1 found, for example, in [PW] can b e used essentially

unchanged to prove the generalization of 1.1 given in 1.7 a b elow. The pro of of

1.7 b is straightforward; the inclusion w s follows from 2.1 of [PV].

1.7 Prop osition. a A completely regular space X; is feebly compact if and

only if it is pseudo compact.

b If X; is any space, then sw =w s .

Following Guthrie and Stone [GS], we will call an expansion of a top ology

on X an R-invariant expansion if C X; =C X; . Theorem 1.9 b elow links

this notion to those of 1.5. We precede it with Lemma 1.8; 1.8 a app ears on

page 103 of [Ca ] and in 1Q of [PW], while 1.8 b may b e found in 1G1 of [GJ].

1.8 Lemma. a Continuous images of feebly compact spaces are feebly compact.

b Continuous images of pseudo compact spaces are pseudo compact.

1.9 Theorem. a If is an expansion of a top ology on a set X , then X;

is feebly compact resp. pseudo compact only if X; is feebly compact resp.

pseudo compact.

b If X; is a space, then is an R-invariant expansion of w and hence

of s , i.e. C X; =C X; s =C X; w ;thus X; is pseudo compact if and

only if X; s is if and only if X; w is.

Clearly 1.9 a follows from 1.8. The rst assertion in 1.9 b follows from 1.7 b

and a sp ecial case of a result attributed to Katetov on page 4 of [GS]; a pro of of

it may b e found by combining 2.2g2 of [PW] and 1.7 b. The second part of

1.9 b clearly follows from the rst.

Finally, in [Ca ]itisshown that: 3

130 J.R. Porter, R.M. Stephenson Jr., R.G. Woods

1.10 Prop osition. A maximal pseudo compact space is T .

In what follows, all hyp othesized separation axioms will be stated explicitly.

Unde ned concepts and notation are explained in [GJ] and/or [PW]. In particular,

N denotes the set of p ositiveintegers.

x2A characterization of maximal pseudo compactness

In this section wecharacterize maximal pseudo compact spaces Theorem 2.4,

and derive some consequences of this characterization Corollaries 2.5 and 2.6.

After intro ducing the concept of a \Herrlich expansion" of a top ology, and deriving

some prop erties of it 2.7 and 2.8, we use such expansions to show in 2.10 that the

statement obtained from 1.6 by replacing \feebly compact" by \pseudo compact"

throughout is in fact false, and that maximal pseudo compactness and maximal

feeble compactness do not have parallel characterizations.

The following de nition app ears in [M]; Theorem 2.2 combines Theorem 2.1

and Corollary 1 of [M].

2.1 De nition. A subset S of a space X is called relatively pseudocompact if

for all f 2 C X , f jS 2 C S .

2.2 Theorem. Let X b e a space.

a If C 2 coz X and if cl C is a relatively pseudo compact subset of X , then

cl C is a pseudo compact subspace of X .

b If X is pseudo compact and C 2 coz X , then cl X is pseudo compact.

We can generalize 2.2 as follows. The pro of b elow is mo delled after Mandelker's

pro of of 2.2.

2.3 Theorem. Let X; b e a space and let V b e a nonempty union of cozero-

sets of X; . Then:

a If cl V is a relatively pseudo compact subset of X , then it is a pseudo-

compact subspace of X .

b If X; is pseudo compact, then cl V is pseudo compact.

Proof: Supp ose that cl V is not a pseudo compact subspace of X . We will show

that cl V is not a relatively pseudo compact subset of X; .

If f 2 C cl V n C cl V , there exists d V such that f d

i i+1

i2N

f d +1. We may assume without loss of generality that f 0. As V is

union of cozero-sets of X; , for each i 2 N there exists C 2 coz X; such

that d 2 C V . Let C = fC : i 2 Ng. Then C 2 coz X; , C V , and

i i i i

by 1.19 of [GJ] the set d = E is C -emb edded in cl V and hence in cl C .

i2N

Nowcl C n C is a zero-set of cl C ,soby 1.18 of [GJ] there exists g 2 C cl V

such that 0 g 1, g d = 1 for each i 2 N, and g [cl C n C ]=f0g. De ne

h: X ! R by: h[X n C ]=f0g, hjcl C = fg. Then h 2 C X; by 1A1 of [GJ],

and h[d ]isunb ounded, so hjcl V 2 C cl V n C cl V . Hence cl V is

i2N

not a relatively pseudo compact subset of X .

Maximal pseudo compact spaces 131

b Since X is pseudo compact, cl V is relatively pseudo compact.

Wenowcharacterize maximal pseudo compact spaces.

2.4 Theorem. Let X; b e a pseudo compact space. The following are equiva-

lent.

a X; is maximal pseudo compact.

b If F X and F is a relatively pseudo compact subspace of X; X n F ,

then F is closed in X; .

Proof: b implies a: Supp ose a fails. Then there is a prop er pseudo compact

expansion of . Supp ose F X and X n F 2 n . Then 6= X n F , and

by 1.9 a X; X n F is pseudo compact. Hence F is a relatively pseudo compact

subset of X; X n F but is not closed in X; . Hence b fails.

a implies b: Supp ose that a holds but b is false. Then there exists

F X such that X n F 2= but F is a relatively pseudo compact subset of

X; X n F . Let p 2 cl F n F and put H =cl F nfpg. Then X n H 2= ,soby

the maximalityof the space X; X n H is not pseudo compact. Hence there

exists f 2 C X; X n H for which f [X ] has no upp er b ound in R.

Clearly X n H jX nfpg = jX nfpg. Since X nfpg 2 as X; is T

by 1.10, it follows that f is -continuous, and hence X n F -continuous, at

each p oint of X nfpg. Let W be a neighborhood of f p in R. Since f is

X n H -continuous at p, there exists V 2 such that p 2 V and f [V n H ] W .

Since X; is maximal pseudo compact it is submaximal by 1.4 a and hence by

1.4 c cl F n F is a discrete subspace of X; . Hence there exists T 2 such

that T \ cl F n F = fpg. Thus T \ X n H = T \ X n F , and so the set

U = V \ T \ X n F isa X n F -neighb orho o d of p for which f [U ] W . Thus

f is X n F -continuous at p, and so f 2 C X; X n F .

As F is a relatively pseudo compact subspace of X; X n F , there exists

k > 0 such that f [F ] k; k. As H = cl F , it follows that f [H ]

X nH

[k; k]. Cho ose b 2 R such that b>maxfk; f pg. De ne g : X ! R as follows:

g x = max f x;b . Since f is -continuous at each p ointofX nfpg,sois g ;as

f [X ]isunb ounded so is g [X ]. Since f is X n H -continuous at p, there exist

J;K 2 such that p 2 J [ K n H and f [J [ K n H ] f p 1;b . Clearly

g [J [ K n H ] = fbg by the de nition of g , and as f [H ] [k; k], it follows

that g [J [ K ] = fbg. As J [ K is a -neighborhood of p, it follows that g is

-continuous at p. Thus g 2 C X; n C X; , contradicting the assumption

that X; is pseudo compact. Hence assuming that a holds and b is false

leads to a contradiction, and so a implies b.

2.5 Corollary. a If X; is maximal pseudo compact then it is submaximal

and its pseudo compact subspaces are closed and maximal pseudo compact.

b If X; is a pseudo compact submaximal space whose pseudo compact sub-

spaces are closed, then its pseudo compact subspaces are of the form A [ T , where

132 J.R. Porter, R.M. Stephenson Jr., R.G. Woods

A 2 R and T is nite. In particular, this holds for maximal pseudo compact

spaces.

Proof: a Submaximality follows from 1.4 a. Let F be a pseudo compact

subspace of the maximal pseudo compact space X; . As X n F jF = jF it

follows that if f 2 C X; X n F then f jF 2 C F; jF and so f [F ] is a b ounded

subset of R since F; jF is pseudo compact. Hence F; X n F jF is a relatively

pseudo compact subspace of X; X n F . Thus by 2.4 F is a closed subspace of

X; .

To show that F; jF is maximal pseudo compact, supp ose that G F and

that G is a relatively pseudo compact subspace of F; jF F n G . Now jF F n

G= X n G jF ,soiff 2 C X; X n G , then f jF 2 C F; jF F n G .

Consequently f [G] is a b ounded subset of R, and so G is a relatively pseudo com-

pact subspace of X; X n G . By 2.4 and the maximal pseudo compactness of

X; it follows that G is closed in X; and hence in F; jF . Thus by 2.4

F; jF is maximal pseudo compact.

b Let F b e a pseudo compact and hence closed subspace of the space X; ,

whose prop erties are as hyp othesized. Then F = cl int F [ T , where T is

de ned to be F n cl int F . As X; is submaximal, by 1.4 c T is a closed

discrete subspace of it. But T is op en in F , so T is an op en-and-closed set of

isolated p oints of F . If T were in nite it would fail to be pseudo compact, in

contradiction to 2.2 b. Hence T is nite.

2.6 Corollary. If X is maximal pseudo compact and if V is a union of cozero-sets

of X then cl V is a maximal pseudo compact subspace of X .

Proof: This follows from 2.2 a, 2.3 a and 2.5.

In analogy with the characterization of maximal feebly compact spaces given

in 1.6, one might conjecture that a pseudo compact space is maximal pseudo com-

pact if and only if it is submaximal and its pseudo compact subspaces are closed.

As 2.5 a attests, maximal pseudo compactness implies submaximality and that

pseudo compact subspaces are closed; however, the converse fails. The remainder

of this section is devoted to a discussion of \Herrlich expansions" of top ologies,

and how the Herrlich expansion of the closed unit interval for example witnesses

the failure of the ab ove-mentioned converse.

2.7 De nition. A Herrlich expansion of a top ology on a set X is a top ology

generated by [fA; B g, where fA; B ; C g is a partition of X into three subsets

each of which is dense in X; . As indicated in the intro duction, we denote such

a top ology as A; B .

Herrlich expansions were considered by Herrlich in [Her], and are also discussed

in Stephenson [St ]. Note that A; B =fS [ T \ A [ U \ B :S; T ; U 2 g. We

will make use of the following prop erties of Herrlich expansions. These prop erties

are due to Herrlich, and pro ofs are at least implicit in [Her], but for the sakeof

completeness wesketch pro ofs here.

Maximal pseudo compact spaces 133

2.8 Prop osition. Let X; b e a space and fA; B ; C g a partition of X into three

dense subsets of X; . Denote A; B by . Then:

a C X; =C X; and hence w = w . See the paragraph following 1.4

for notation.

b A; B 2 RO .

c If is a semiregular top ology then so is .

d If some p ointofA [ B is a regular G -p ointofX; , then X; is not

feebly compact. A p oint p 2 X is called a regular G -p ointofX; if

T T

there exist V such that fpg = fV : n 2 Ng = fcl V :

n n n

n2N

n 2 Ng.

Proof: Sketch: One can prove that:

i If T 2 then cl T \ A = cl T n B and cl T \ B = cl T n A.

ii If T 2 then cl T =cl T .

a Since , clearly C X; C X; . Conversely, supp ose that f 2

C X; . Clearly f is -continuous at each p ointofC . If x 2 A and f x 2 V ,

0 0

where V is op en in R, nd op en subsets U and Y of R such that f x 2 U

cl U Y cl Y V . As f 2 C X; , there exists T 2 such that x 2 T

R R

and f [T \ A] U . Thus f [cl T \ A] cl f [T \ A] cl U Y . But

R R

T \ C cl T \ A by i, so f [T \ C ] Y . As f is -continuous at each

p ointofC , for each x 2 T \ C there exists W x 2 such that x 2 W x and

f [W x] Y . Let W = [W x:x 2 T \ C ]; then T \ C W and f [W ] Y .

Now cl T = cl T \ C as C is -dense in X . But cl T \ C cl W and

cl W = cl W by ii ab ove; hence cl T cl W , and as f is -continuous,

f [cl T ] f [cl W ] cl f [W ] cl Y . Thus f [T ] V ,sof is -continuous at

R R

x . Interchange the roles of A and B in the ab ove and conclude that f 2 C X; .

b Setting T = X in i ab ove, we see that B = X n cl A and A = X n cl B .

Thus A; B 2 RO .

c By hyp othesis fS [ T \ A [ W \ B :S; T ; W 2 RO g is a base for .

By i ab ove,

int cl T \ A=int [cl T \ X n B ]

=int cl T \ int X n B :

But int cl T =int cl T by ii ab ove and int X n B =A by i ab ove. Thus

since T 2 RO , we see that int cl T \ A=T \ A. Similarly W \ B 2 RO ,

and S 2 RO by ii ab ove. The result follows.

T T

d Supp ose p 2 A and fpg = fV : n 2 Ng = fcl V : N 2 Ng, where

n n

V is, without loss of generality, a decreasing sequence of memb ers of .

n2N

Then fV \ B : n 2 Ng is an in nite, lo cally nite in X; subfamily of .

Hence X; is not feebly compact.

134 J.R. Porter, R.M. Stephenson Jr., R.G. Woods

2.9 Lemma. Let and be two top ologies on a set X . If s , then

s = s .

Proof: Let U 2 . Then U 2 and cl U cl U cl U . As int cl U 2 ,

by 2.2 f of [PW] it follows that cl U =cl U =cl U . Thus X n cl U 2 , and

by rep eating the ab ove calculation with X n cl U in place of U and then taking

complements, we see that int cl U =int cl U . It follows that s s .

Rep eating the ab ove with replacing and s replacing we see that ss

s . By 2.2 f 6 of [PW] we see that ss =s , and so s s . Hence s = s .

The following technical prop erty of maximal pseudo compact expansions of Her-

rlich expansions will b e key in the construction of examples.

2.10 Lemma. Let X; b e a semiregular pseudo compact space, and let b e the

Herrlich expansion of generated by [fA; B g, where fA; B ; C g is a partition of

X into three dense subsets of X; . If is a maximal pseudo compact expansion

of , then s n 6= ;. See remarks preceding 1.5 for notation.

Proof: As X; is semiregular, so is X; see 2.8 c. If the lemma fails

then s , and hence by 2.9 ab ove it follows that s = s = . Thus

as A 2 RO =RO , we see that cl A =cl A = A [ C as noted in i in

the pro of of 2.8. Let p 2 C and set E = B [fpg; by the ab ove, B [fpg 2= ,

so E is a prop er expansion of . We will show that C X; =C X; E ,

which will imply that X; E is pseudo compact and thereby contradict the

maximality of . Let f 2 C X; E . By 1.10 X nfpg 2 . Observe that

jX nfpg = E jX nfpg, and hence f is -continuous at each p ointofX nfpg.

We will show that f is -continuous at p, thereby showing that f 2 C X; and

completing our pro of.

Let W b e an op en subset of R for which f p 2 W . Cho ose op en subsets U

and V of R such that f p 2 U cl U V cl V W . By 1.4 a and

R R

1.5 e there is an ultra lter V of dense subsets of X; such that = V .

Since f is E -continuous at p, there exist D 2 V and T 2 such that p 2

S = D \ T \ B [fpg and f [S ] U . Nowas p 2= A [ B and = A; B , we

may assume that T 2 . Because p 2 S , it follows that cl S =cl S . Since

= V , the -dense subsets of X are precisely the memb ers of V . Thus D is

a dense subset of X; and so cl D \ T \ B =cl T \ B =cl T \ B =

cl T \ B T \ B [ C . Consequently

f [T \ B [ C ] f [cl D \ T \ B ] f [cl S ]=f [cl S ]

cl f [S ] cl U V:

R R

Also note that since E \ A = ;, it follows that if L X then

ii A \ cl L = A \ cl L:

Maximal pseudo compact spaces 135

In particular,

iii A \ cl T \ A \ f [V ] = A \ cl T \ A \ f [V ]:

We claim that T \ A \ f [V ] is E -dense in T \ A. If not, then by iii

ab ove there would exist D 2 V and P 2 such that P \ D \ T \ A 6= ; but

P \ D \ T \ A \ f [V ]=;. Let P \ T = G; then f [G \ D \ A] R n V , and as

f is E -continuous it follows that f [cl G \ D \ A] R n V . But by ii

ab ove,

A \ cl G \ D \ A=A \ cl G \ D \ A

G \ A;

the latter inclusion follows from the fact that D 2V and hence is dense in X; ,

while G \ A 2 . Hence f [G \ A] R n V . Now G 2 n f;g so G \ C is in nite.

Hence we can cho ose q 2 G \ C nfpg. By i in the pro of of 2.8, q 2 cl G \ A.

But cl G \ A=cl G \ Aas = s and G \ A 2 . Hence q 2 cl G \ A.

If q 2 J [ K \ E , where J;K 2 then q 2 J as q 2= E so J \ G \ A 6= ;.

Hence basic E -neighborhoods of q meet G \ A, and so q 2 cl G \ A.

Consequently

f q 2 f [cl G \ A] cl f [G \ A] R n V see ab ove:

But q 2 G \ C T \ B [ C , so by i f q 2 V . This contradiction veri es our

claim that T \ A \ f [V ]is E -dense in T \ A.

Thus

f [T \ A] f cl T \ A \ f [V ]

cl V W:

Thus f [T ] = f [T \ B [ C ] [ f [T \ A] V [ W W . Consequently f is

-continuous at p, and so as noted ab ove, f 2 C X; . Thus C X; E =

C X; , which as observed ab ove violates the maximality of . The lemma

follows.

The following class of examples shows that pseudo compact submaximal spaces

whose pseudo compact subspaces are closed in fact, whose singleton sets are zero-

sets need not b e maximal pseudo compact.

2.11 Example. Let X; b e a semiregular pseudo compact space whose pseudo-

compact subspaces are closed; supp ose also has a Herrlich expansion . These

conditions will be satis ed if X; is a compact metric space without isolated

p oints, for example. Let V be a dense ultra lter on X; . As V is an ex-

pansion of , by 1.8 b and hyp othesis, pseudo compact subspaces of X; V

are closed. Furthermore, by 1.5 a X; V is submaximal and by 1.5 b and

136 J.R. Porter, R.M. Stephenson Jr., R.G. Woods

1.9 b it is pseudo compact. But since s V =s by 1.5 b and s = by 2.8 c

it follows that s V = , and hence X; V cannot b e maximal pseudo compact

by 2.10.

Observe also that b oth X; and X; V are pseudo compact spaces that

have regular closed subspaces that fail to b e pseudo compact spaces. This con-

trasts to feebly compact spaces, whose regular closed subspaces are feebly com-

pact. If = A; B , where fA; B ; C g is a partition of X into three dense

subspaces of X; , then cl A = A [ C see i in pro of of 2.8 and as A [ C

fails to b e closed in X; , it is not a pseudo compact subspace of X; . Hence

cl A is a regular closed subset of X; whichby 1.8 b is not a pseudo compact

subspace of X; . By 1.5 b cl A is a regular closed subspace of X; V and

by 1.8 b is not a pseudo compact subspace of it.

x3 Construction and examples of maximal pseudo compact spaces

Having characterized maximal pseudo compact spaces in 2.4, and shown in

2.10 that a plausible alternate candidate for such a characterization fails, we

now consider ways of constructing maximal pseudo compact spaces. We b egin by

investigating maximal pseudo compact expansions of pseudo compact Tychono

space whose pseudo compact subspaces are closed.

The following de nition rst app eared in [W].

3.1 De nition. Let T X denote the set of top ologies on a nonempty set X . We

de ne on T X as follows. if and for each V 2 ,cl V =cl V .

Clearly is a partial order on T X , and s for each 2 T X by

1.5 b.

3.2 Lemma. Let X; be a pseudo compact Tychono space in which every

pseudo compact subspace is closed. If is a pseudo compact expansion of , then

w = and .

Proof: As and X; is Tychono , it follows that = w w . Now

w is a Tychono top ology as is an expansion of a Tychono top ology. If

A 2 R w , then by 2.3 b A is a pseudo compact subspace of X; w . By

1.8 b A is a pseudo compact subspace of X; and hence is closed in X; .

Hence sw . Since w is Tychono , it follows that sw = w , and so

= w .

If fails, there exists V 2 for whichcl V 6=cl V . Nowcl V cl V

since , so there exists p 2 cl V n cl V . Let F =cl V nfpg; then X n F 2= so

as pseudo compact subsets of X; are closed, F is not a pseudo compact subspace

of X; . Hence F is not a feebly compact subspace of X; , and so there is

a countably in nite pairwise disjoint family M of nonempty op en subsets

n2N

of F; jF that is lo cally nite in F; jF . As V is -dense in F , it follows that

M \ V is a countably in nite family of nonempty memb ers of that is

n2N

lo cally nite in F; jF . Denote M \ V by L and let y 2 L . As X;

n n n n

is Tychono there exists g 2 C X; such that 0 g 1, y 2 int g 1

n n n n

Maximal pseudo compact spaces 137

and g [X n L ] = f0g for each n 2 N. Let f = fng : n 2 Ng. Clearly

n n n

f is well-de ned as for each x 2 X , g x 6= 0 for only nitely many n 2 N

recall M is pairwise disjoint. As L is a lo cally nite subfamily

n n

n2N n2N

of F; jF , clearly f jF 2 C F; jF . But f is identically zero on X n V , so

f jX n V 2 C X n V; jX n V . Consequently f jX nfpg2 C X nfpg;jX nfpg.

But as p 2= cl V , and f [X n cl V ] = f0g, it follows that f is continuous at p

with resp ect to the top ology . But as f jX nfpg 2 C X nfpg;jX nfpg

C X nfpg;jX nfpg, it follows that f 2 C X; . As X; is pseudo compact

and f is unb ounded, this is a contradiction. Consequently as claimed.

3.3 Theorem. Let X; be a pseudo compact Tychono space in which each

pseudo compact subspace is closed. Then:

a If V is a dense ultra lter on X; , then X; V is a maximal pseudo-

compact space.

b If is a maximal pseudo compact expansion of that is feebly compact,

then there is a dense ultra lter V on X; such that = V and hence

= s .

Proof: a As X; isTychono and pseudo compact, by 1.1 it is feebly com-

pact. By 1.5 c, V is also a feebly compact top ology on X . Thus by 1.2

X; V is pseudo compact. Now supp ose that were a pseudo compact expan-

sion of V and hence of . Wenow can argue as in the pro of of Theorem 11 of

[GS]; what follows is an elab oration and slight mo di cation of the brief argument

presented there, where X; is also assumed to b e rst countable. Denote V

by , and supp ose that U 2 n . By 3.2 w = and ; hence by the

equivalence of 1 and 3 in Theorem 6 of [GS], there exist V 2 and D X

such that U = V \ D and D is -dense in X . Without loss of generality we

may assume that X n V D . Replace D by D [ X n V if necessary. Now

U 2= so D 2= V ; consequently by the maximality of V , there exists E 2 V

such that D \ E is not -dense in X . Hence there exists W 2 n f;g such that

W \ D \ E = ;. Let T = W \ V . Then T 6= ;, for if T = ; then W X n V

and hence W D . Consequently ; = W \ D \ E = W \ E , which is a con-

tradiction as E is -dense and W 2 n f;g. As T 2 and U 2 , it follows

that T \ U 2 . But T \ U =W \ V \ V \ D =W \ V \ D ,soT \ U 6= ;

since W \ V 2 n f;g. However, T \ U \ E =W \ D \ E \ V = ;. Cho ose

p 2 T \ U ; then E [fpg2 V since E 2V, so E [fpg2 . Consequently

T \ U \ E [fpg 2 ;however, T \ U \ E [fpg= fpg,sofpg2 . Thus

the characteristic function 2 C X; , but as fpg 2= since E is -dense

fpg

and p 2 X n E , 2= C X; . But w = by 3.2, and this implies that

fpg

C X; = C X; . Hence we have reached a contradiction, and so n = ;.

Hence V has no prop er pseudo compact expansions, and hence it is a maximal

pseudo compact top ology on X .

b If were a feebly compact expansion of , then by 1.2 would b e a pseu-

do compact expansion of . Consequently = by the maximalityof . Thus

is a maximal feebly compact expansion of . In [PSW ] itisshown that if each 2

138 J.R. Porter, R.M. Stephenson Jr., R.G. Woods

feebly compact subspace of X; is closed, then each maximal feebly compact

expansion of X; is of the form V , where V is a dense ultra lter on X; .

Now subspaces of Tychono spaces are feebly compact iff they are pseudo com-

pact, so it follows that = V for some dense ultra lter V on X; . Then

= s by 1.5 b.

The class of examples presented in 2.11 witnesses the fact that the assumption

that X; is Tychono cannot be dropp ed in 3.3 a. A Herrlich expansion

of the interval top ology on the closed unit interval is a Urysohn, semiregular

pseudo compact top ology, and pseudo compact subspaces of [0; 1]; are closed,

but no top ology on [0; 1] of the form V is maximal pseudo compact where V is

a dense ultra lter on [0; 1]; .

Note that the sp ecial case of 3.3 a in which the hyp othesis \each pseudo com-

pact subspace is closed" is replaced by the stronger hyp othesis \X; is rst

countable" was proved in Corollary 11 a of [GS].

Finally, itisshown in [PSW ] that the Tychono plank T; do es not have

a maximal feebly compact expansion. If V were a dense ultra lter on T; , then

T; V would have a dense set of isolated p oints as T; do es. In 4.2 we will

show that a space with a dense set of isolated p oints is maximal feebly compact

if and only if it is maximal pseudo compact. Consequently no expansion of

of the form V can be a maximal pseudo compact top ology. This shows that

the hyp othesis \each pseudo compact subspace is closed" cannot b e dropp ed from

3.3 a.

We give another sucient condition for maximal pseudo compactness. Recall

that a space X; is called R-maximal if C X; n C X; 6= ; whenever is

a prop er expansion of . This concept is discussed in [GS] in some detail.

3.4 Theorem. Supp ose that X; is an R-maximal pseudo compact space such

that every pseudo compact subspace of X; w is a closed subspace of X; .

Then X; is a maximal pseudo compact space.

Proof: Supp ose that is a prop er expansion of and that X; is pseudo-

compact. Then by 1.8 b X; w is pseudo compact and hence feebly compact

by 1.7 a. Hence every regular closed subset of X; w is feebly compact by

1.4 a of [PSW ], and hence is pseudo compact by 1.2. Now so w w ;

consequently every regular closed subset of X; w is a pseudo compact subspace

of X; w by 1.8 b and hence byhyp othesis is closed in X; . But X; w is

semiregular by 1.7 b, so every closed subset of X; w is closed in X; . Conse-

quently w w , and hence C X; =C X; w . But C X; w =C X;

and by the R-maximalityofX; it follows that = . Hence X; is maximal

pseudo compact.

3.5 Corollary. If X; is a pseudo compact Tychono space whose pseudo com-

pact subspaces are closed, and if is a pseudo compact R-maximal expansion of

, then X; is a maximal pseudo compact space.

The next example shows that the assumption in 3.2 that every pseudo compact

Maximal pseudo compact spaces 139

subset of X; be closed cannot b e dropp ed, and that in contrast to 3.3 not

every maximal pseudo compact completely Hausdor i.e. distinct p oints can b e

separated by a real-valued continuous function space X; has the prop erty

that each pseudo compact subspace of X; w is a closed subspace of X; w .

Recall that if M is maximal almost disjoint henceforth abbreviated m.a.d.

family of in nite subsets of N, and if fpM : M 2Mg is a set D , disjoint from

N, faithfully indexed by M, then the set Y = N [ D can be given a top ology

M as follows:

M=fU Y : if pM 2 U then M n U is niteg:

The resulting space Y; M will b e denoted by M. It is a lo cally compact

pseudo compact Hausdor space; its set of isolated p oints is N and is dense, and

D is an uncountable closed discrete subset of it. See 5I of [GJ] or 1N of [PW] for

details.

Observe that if U 2 M and [cl U \ N] n N is in nite, then it is un-

countable; for by 1Q2 and 1.11 d2 of [PW], cl U \ N is pseudo compact,

but it is not countably compact as [cl U \ N] n N is an in nite closed discrete

subset of it. Consequently it cannot b e countable see 5F5 of [PW].

Also observe that if W is op en in M and D n W is nite, then M n W

is compact. To see this supp ose D n W = fpM :i =1 to ng, where fM : i =1

i i

fM : i = 1 to ng n W is nite; for if not, by the to ng M. Then N n

T S

maximalityof M there exists S 2Msuch that S [N n fM : i =1tong n W ]

is in nite. One easily checks that pS 2 D n W but pS 2f= pM : i =1to

ng, which is a contradiction. Thus N n fM : i =1 to ng n W is nite, and

S S

M [fpM g : i =1 to n [ N n fM : i =1 to ng n W M n W

i i i

which is compact. Thus M n W is compact.

In Theorem 1 of [Hec] it is shown that there exists a m.a.d. family M on N

and a subset B of D = M n N with this prop erty:

jB j = c and if U 2 M and jU \ B j > @ then jcl U \ D n B j = c

Here c denotes 2 . Observe that implies that jD n B j = c.

If B is partitioned into two sets A and A , each of cardinality c, then clearly

1 2

remains true if B is replaced by either A or A . Henceforth we assume that

1 2

M is as in , and that A and A are as ab ove. We let M n N = D , and

1 2

D n A [ A =E .

1 2

3.6 Lemma. Let M be as ab ove, let f 2 C M;M , and let r > 0 and

">0 b e given. If there is a nite subset G of E [ A such that f [E [ A n G]

1 1

[r;r], then there exists a nite subset F of E [ A such that f [E [ A n F ]

" "

2 2

[r "; r + "].

Proof: Supp ose not; then there exists an in nite subset S of E [ A such

that jf xj > r + " whenever x 2 S . By hyp othesis f y 2 [r;r] for all but

140 J.R. Porter, R.M. Stephenson Jr., R.G. Woods

nitely many memb ers y of E , so without loss of generality,wemay assume that

g. Clearly S cl U \ N n S A . Let U = fx 2 M:jf xj >r+

N, so cl U \ N is in nite and hence, as noted ab ove, uncountable. But

cl U \ N n N fx 2 M:jf xj >r+ g, which is an op en set of M

that we denote by V . Since byhyp othesis f takes all but nitely many memb ers

of E [ A into [r;r], clearly jV \ A j > @ . But as holds with B replaced by

1 2 0

= c. Thus there are c elements y of A , it follows that cl V \ E [ A

2 1

, contradicting the ab ove-mentioned hyp othesis. E [ A for which jf y jr +

The lemma follows.

Observe that 3.6 holds if A and A are interchanged. The example X; con-

1 2

structed b elow is a mo di cation by Hechler [Hec] of an example due to Stephen-

son [St ]. We mo dify it further to construct an example showing that 3.2 cannot

b e generalized much.

3.7 Example. . Let M, A , A , D and E b e de ned as ab ove, and note that

1 2

as jE j = jA j = jA j = c, there are bijections j : A ! E i = 1; 2. De ne

1 2

i i

a partition P on M N.

n o

P = j d;n 1 ; d; n; j d;n +1 : n 2 N;n > 1; and d 2 E :

1 2

n o

[

x; 1; j x; 2 : x 2 E

n o

[ [

P = P fxg : x 2 M N and x 2= fL : L 2P g :

1 1

Let q : M N !P map each p ointof M N to the member of P to which

it b elongs, give P the quotient top ology induced by q , and let X = P [ f1g,

where 1 2= P . Top ologize X by letting P ; be an op en subspace of X and

letting f1g [ fT : n k g : k 2 N be a neighb orho o d base at 1, where

T is de ned to be q [ M fng]. Denote the resulting top ology on X by .

One veri es easily that X; is a rst countable T space with a countable dense

set of isolated p oints namely N N. Observe that each T is homeomorphic

to M. Note that X; can be intuitively visualized as b eing obtained by

gluing for n>1 the nth copy T of M to the n 1st and n + 1st copies

by gluing E in the nth copy to A in the n 1st copy and to A in the

1 2

n + 1st copy; the copyofE in T is also glued to the copyof A in T . Then

1 2 2

the sequence T of glued copies is required to converge to 1.

n2N

Observe that X; is pseudo compact; for if f 2 C X; and f 1=r , then

S S

there exists n 2 N such that fT : k ng f [r 1;r + 1]. But fT : k

k k

ng is a nite union of pseudo compact spaces and hence is pseudo compact; thus f

is b ounded on this union and hence on X; .

It is easily veri ed that X n f1g;jX n f1g is a lo cally compact zero-

dimensional Hausdor space, and its one-p oint compacti cation can be viewed

as X equipp ed with a top ology ; as is a Tychono top ology, it is evident

Maximal pseudo compact spaces 141

that w . We claim that = w ; to prove this it clearly suces

to show that if f 2 C X; , f 1 = 0, and r > 0, then X n f [r;r]

r r

; ]. As is a compact subspace of X; . To verify this, let U = f [

2 2

noted ab ove, there exists k 2 N such that fT : m k g U . Then

r r

q [E [ A fk g] f [ ; ]. Applying 3.6 with M replaced by its

2 2

r r

, and G by ;we see that there is a nite subset homeomorph T , r by , " by

r r r r

fk g]] [ ; + ]. But F of E [ A such that f [q [ E [ A n F

2 2

k k

2 2

8k 8k

q E [ A n F fk g = q E [ A n F fk 1g by de nition of q ,sowe

2 1

k k

r r r

can apply 3.6 again, this time with M replaced by T , by + , and G

k 1

2 2

by F . We continue this pro cess \inductively downward" until we reach T . We

r r r r

see that at each stage j = k to 1 f [ ; + ] is an op en set containing

2 4 2 4

all p oints of q [D fj g] except p erhaps for the nite set F ; hence by the remark

b efore our de nition of ,

i h

r r r r

; + is a compact subset K of T : T n f

j j j

2 4k 2 4k

Thus X n f [r;r] fK : j =1tok g, which is compact, and our veri cation

is complete.

Observe that q [ M f1g] is a pseudo compact subspace of X; w that is

not closed there. Furthermore, the unique dense ultra lter on X; isfJ S :

q [N N] J g and by 2.25 of [PSW ], X; V is maximal feebly compact and

hence by 4.2 b elow maximal pseudo compact. Nowif V = N f1g, then V 2 w

but one can show that 12cl V n cl V ; consequently it is false that w .

Using S; w in place of X; , this shows that 3.2 can fail if the assumption that

pseudo compact subspaces b e closed is dropp ed. Similarly, since w V = w ,

X; V is maximal pseudo compact but not all pseudo compact subspaces of

Xw V are closed in X; w V .

x4 Maximal pseudo compactness vs. maximal feeble compactness

In this section weinvestigate the relationship b etween maximal feebly compact

spaces and maximal pseudo compact spaces. In 4.1 and 4.2 we present classes of

spaces for which the concepts are equivalent. In 4.3 and 4.4 we present su-

cient conditions for a maximal feebly compact space to b e maximal pseudo com-

pact. Since maximal feeble compactness has an easily applicable characterization

see 1.6, these criteria are useful. In 4.5 we present a \structure theorem" com-

mon to maximal feeble compactness and maximal pseudo compactness. Finally,

in 4.6 we give an example of a maximal feebly compact space that is not maxi-

mal pseudo compact. Wehave b een unable to determine whether every maximal

pseudo compact space must b e maximal feebly compact.

In [Ca ], Cameron proved that a Tychono space is maximal feebly compact

if and only if it is maximal pseudo compact. The following result generalizes this.

Recall that a subfamily F of n f;g is a -base for the space X; if each

nonempty op en subset has a subset b elonging to F . See [Ho] or 2N5 of [PW] for details.

142 J.R. Porter, R.M. Stephenson Jr., R.G. Woods

4.1 Theorem. a If X; is a maximal feebly compact space, and if coz X; n

f;g is a -base for , then X; is maximal pseudo compact.

b If X; is a maximal pseudo compact space that is feebly compact, then

it is maximal feebly compact.

c If X; is a maximal pseudo compact space, and if coz X; nf;g is a -base

for s , then X; is maximal feebly compact. [Observe that coz X; s .]

Proof: a If X; were maximal feebly compact, then by 1.2 it would be

pseudo compact. Supp ose that were a pseudo compact expansion of . We will

show that X; is feebly compact; by the maximalityof , this will imply that

= and hence that is a maximal pseudo compact top ology.

If X; were not feebly compact, there would exist a pairwise disjoint family

V n f;g with no limit p ointinX; . Let A = fn 2 N :int V = ;g.

n n

n2N

By 1.4 b X; is submaximal, and hence V is a discrete subspace of X;

for each n 2 A. Thus V consists of isolated p oints of X; as V 2 and

n n

. Cho ose x 2 V for each n 2 A; then fx g has no limit p oint

n n n

n2A

in X; as V has none. Hence fx g is an op en-and-closed discrete

n n

n2A

n2N

subset of X; . De ne f : X ! R by: f x =n, f [X nfx : n 2 Ag]=f0g;

n n

then f 2 C X; . If A were in nite then f would b e unb ounded, contradicting

the assumption that X; is pseudo compact. Thus A is nite. Consequently

int V 6= ; for co nitely many n 2 N; without loss of generality assume that

int V 6= ; for all n 2 N. By hyp othesis there exists C 2 coz X; n f;g such

n n

that C V . NowC is lo cally nite in X; as V is. Cho ose

n n n n

n2N n2N

p 2 C and f 2 C X; such that 0 f 1, f [X n C ]=f0g, and f p =

n n n n n n n n

1 for each n 2 N. By the lo cal niteness of C the function fnf :

n n

n2N

n 2 Ng is a well-de ned unb ounded member of C X; , again contradicting the

pseudo compactness of X; . Consequently X; must b e feebly compact, and

our result follows.

b If is a feebly compact expansion of , then X; is pseudo compact by

1.2. Hence = by the maximalityof . As X; is feebly compact, it must b e

maximal feebly compact.

c Let X; be as in the hyp othesis of c; we claim it is feebly compact.

If not, there exists V n f;g with no limit p oint in X; . Then

n2N

int cl V also has no limit p oint in X; . By hyp othesis there exists

n2N

C 2 coz X; n f;g such that C int cl V . Arguing as in a, we construct

n n n

an unb ounded member of C X; , contradicting the pseudo compactness of X; .

Hence X; is feebly compact, and c now follows from b.

4.2 Corollary. If X; has a dense set I of isolated p oints, then it is maximal

feebly compact if and only if it is maximal pseudo compact.

Proof: Clearly fxg : x 2 I coz X; n f;g, and byhyp othesis fxg : x 2 I

is a -base for and hence for s . Now apply 4.1.

In 2.16 of [PSW ] we have shown that semiregular maximal feebly compact

spaces must have a dense set of isolated p oints. Clearly they must also b e maximal

Maximal pseudo compact spaces 143

pseudo compact by 4.2.

We now give sucient conditions for a maximal feebly compact space to be

maximal pseudo compact. Theorem 4.3 b elow complements 3.3 b.

4.3 Theorem. Let X; b e a pseudo compact Tychono space whose pseudo-

compact subspaces are closed. If is a maximal feebly compact expansion of ,

then is maximal pseudo compact.

Proof: Since each feebly compact subspace of X; is closed by 1.1 and hy-

p othesis, it follows as in the pro of of 3.3 b that each maximal feebly compact

expansion of is of the form V , where V is a dense ultra lter on X; . Our

result now follows from 3.3 a.

4.4 Corollary. If X; is a maximal feebly compact space whose singleton sets

are zero-sets, then it is maximal pseudo compact.

Proof: The weak top ology w has coz X; w as a base, and it is T as zero-

sets of X; are closed in w . Hence X; w isTychono . As its singleton sets

are intersections of countably many closed neighb orho o ds in X; , it is easy

to prove or, one can app eal to 2.7 c of [PSW ] that its feebly compact and

hence, by 1.1, its pseudo compact subspaces are closed. Hence as is a maximal

feebly compact expansion of w ,by 4.3 X; is maximal pseudo compact.

Wenow consider a prop erty shared by maximal feebly compact and maximal

pseudo compact spaces.

4.5 Prop osition. If X; is a submaximal space without isolated p oints, then

each discrete subspace is closed.

Proof: Let D b e a discrete subspace of X; . As each p ointofint D would

b e isolated in X; , we conclude that int D = ;. By 1.4 c D is closed.

As maximal feebly compact and maximal pseudo compact spaces are submax-

imal, it immediately follows from 4.5 that if such a space is Hausdor and has

no isolated p oints, it contains no nontrivial convergent sequence, and in fact no

in nite countable compact spaces; for if K were such a space, it would contain

an in nite discrete subspace, which by the countable compactness of K would

contain a limit p oint, in contradiction to the ab ove.

We nish this pap er by exhibiting a Hausdor maximal feebly compact space

that is not maximal pseudo compact.

4.6 Example. Let X; denote Bing's example of a countable, rst countable

connected Hausdor space see [Bi]. [The underlying set X is fp; q 2 Q Q :

q 0g, where Q denotes the rationals. A neighb orho o d base at p; 0 is fx; 0 2

g : n 2 N . If p; q 2 X and q>0 let p ; 0 and p ; 0 b e the X : jp xj <

1 2

other two corners of the equilateral triangle in R whose vertex is p; q and whose

base lies on the X -axis. Then a neighb orho o d base at p; q is fp; q g[ fx; 0 :

1 1

jp xj < . Connectedness follows from the or jp xj < g : n 2 N

1 2

n n

144 J.R. Porter, R.M. Stephenson Jr., R.G. Woods

fact that if V; W 2 n f;g then cl V \ cl W 6= ;. It is easy to check that the

semiregularization X; s is also a countable, rst countable connected Hausdor

space. Real-valued continuous functions on connected countable spaces must b e

constant, so X; s is pseudo compact. One can check that int cl fx; 0 :

1 1

4 4

that is lo cally nite in X; s , so X; s is not feebly compact.

By 5.7 and 5.9 of [St ], there exists a rst countable feebly compact Haus-

dor extension T; of X; s . Thus X is a prop er dense subset of T; .

Let V be a dense ultra lter on T; such that X 2 V . Then by 2.25 of

[PSW ], T; V is a maximal feebly compact space. Nowby 2.2i2 of [PW],

s V jX = s V jX . But s V jX = jX = s see 1.5 b, so X; s

is the semiregularization of X; V jX . By 1.9 b X; V jX is pseudo com-

pact. As X 2 V , X; V jX is a dense subspace of T; V , and so not all

pseudo compact subspaces of T; V are closed. Hence by 2.5 a T; V is

not maximal pseudo compact, and hence is an example of a Hausdor maximal

feebly compact space that is not maximal pseudo compact.

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The University of Kansas, Lawrence, Kansas 66045, U.S.A.

The University of South Carolina, Columbia, South Carolina 29208, U.S.A.

The University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada

Received January 18, 1993