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Maximal Pseudocompact Spaces Comment.Math.Univ.Carolinae 35,1 1994127{145 127 Maximal pseudo compact spaces 1 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 1 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 X 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 . 1 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 ]. 4 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 1 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 1 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 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 1 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]. 2 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 . 1 In what follows, all hyp othesized separation axioms will be stated explicitly. Unde ned concepts and notation are explained in [GJ] and/or [PW].
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