summary:The relativization of Gryzlov's theorem about the size of compact $T_1$-spaces with countable pseudocharacter is false

Just, Winfried

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Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeWinfried JustThe sizes of relatively compact T1-spacesCommentationes Mathematicae Universitatis Carolinae, Vol. 37 (1996), No. 2, 379--380Persistent URL: http://dml.cz/dmlcz/118839Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1996Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czComment.Math.Univ.Carolin. 37,2 (1996)379–380 379The sizes of relatively compact T1-spacesWinfried Just1Abstract. The relativization of Gryzlov’s theorem about the size of compact T1-spaceswith countable pseudocharacter is false.Keywords: relatively compact, relatively Lindelöf, T1-space, measurable cardinalClassification: 54A25, 54D20Let X,Y be topological spaces such that Y ⊆ X . We say that Y is rela-tively compact (relatively Lindelöf ) in X if for every open cover U of X thereexists a finite (countable) subfamily V ⊆ U such that V is a cover of Y . The no-tion of relative compactness was introduced (under a slightly different name) byD.V. Ranchin [R]. A.V Arhangel’skii showed that if Y is regular in X (i.e., for eachclosed subset B ofX and each point y ∈ X\B, there is an open neighborhood U ofy in X such that cl(U)∩B = ∅), X is first-countable at each point in Y , and Y isLindelöf in X , then |Y | ≤ 2ℵ0 (see Corollary 6 of [A] for a somewhat stronger re-sult). Thus, Arhangel’skii’s famous theorem on the size of first-countable regularLindelöf spaces generalizes to the context of relative Lindelöfness.In [G], A.A. Gryzlov proved that every compact T1-space of countable pseu-docharacter has cardinality at most 2ℵ0 . The question arises whether this theoremalso generalizes to the context of relative compactness. Theorem 1 below showsthat this is not the case.1. Theorem. Suppose λ is smaller than the first measurable cardinal. Thenthere exist first-countable T1-spaces Y ⊂ X , such that |Y | = λ, |X | = 2λ, and Yis compact in X .Proof: Let λ ≥ ℵ0 be as in the assumption. We let Y be λ itself with the discretetopology. Let Z be the set of all partitions z̄ = (zn)n∈ω of λ into countably manypairwise disjoint sets. The underlying set of the space X will be Y ∪Z. We definea topology on X as follows:- the points in Y are isolated;- the basic open neighborhoods of z̄ ∈ Z are of the form V z̄m =⋃n≥m zn ∪ {z̄}.Clearly, X and Y are first-countable T1-spaces. It remains to show that Y isrelatively compact in X . The latter is equivalent to the following:1Partially supported by NSF grant DMS-9312363.380 W. Just2. Claim. For every function f : Z → ω there exists a finite set F ⊂ Z suchthat |Y \⋃z̄∈F⋃n≥f(z̄) zn| < ℵ0.Proof: Suppose not and let f be a counterexample. For each F ∈ [Z]<ℵ0 define:YF = Y \⋃z̄∈F⋃n≥f(z̄) zn. Then the family {YF : F ∈ [Z]<ℵ0} generates a filterF of subsets of λ with the following properties:(1) F contains no finite subset of λ;(2) for every partition {zn : n ∈ ω} of λ into countable many pairwise disjointsubsets there exists an m ∈ ω such that⋃n≤m zn ∈ F .The existence of such an F leads to a contradiction, since it implies that λis at least as big as the first measurable cardinal. To see the latter, considerthe characteristic function of F , i.e., the function χF : P(λ) → {0, 1} that takesthe value χF (a) = 1 if and only if a ∈ F . If we identify P(λ) andλ{0, 1}with the product topology, then (1) implies that χF is not continuous. On theother hand, if limn→∞ an = a and zn = {ξ ∈ λ : n = min{k ∈ ω : ∀m ≥k (ξ ∈ am ↔ ξ ∈ a)}}, then {zn : n ∈ ω} is a partition of λ. Applying (2) tothis partition, we can see that χF is sequentially continuous. Now a result ofAntonovskii and Chudnovskii [AC, Theorem 1.3] implies that λ is as least as bigas the first measurable cardinal. In Theorem 1, the assumption that λ is smaller than the first measurablecardinal cannot be dropped, since the following relativized version of a well-knownclassical result holds:3. Proposition. Suppose X is a T1-space and ψ(x,X) < κ for every x ∈ X . IfY is relatively Lindelöf in X , then |Y | < κ.Proof: Suppose X,Y form a counterexample. Let µ : P(κ) → {0, 1} be aκ-additive measure that vanishes on singletons, and assume without loss of gen-erality that κ ⊆ Y . For each x ∈ X , choose {Uxξ : ξ < λx < κ} such that Uxξ isan open neighborhood of x and⋂ξ<λxUxξ = {x}. By κ-additivity of µ, we canpick for every x ∈ X a ξx such that µ(κ ∩ Uxξx) = 0. Now let U = {Uxξx: x ∈ X}.Then U is an open cover of X , but if V = [U ]<κ, then µ(⋃V ∩ κ) = 0, hence Vdoes not cover Y . References[A] Arhangel’skii A.V., A generic theorem in the theory of cardinal invariants of topologicalspaces, Comment. Math. Univ. Carolinae 36.2 (1995), 303–325.[AC]Antonovskii M.Y., Chudnovskii D.V., Some questions of general topology and Tikhonovsemifields II, Russian Math. Surveys 31.3 (1976), 69–128.[G] Gryzlov A.A., Two theorems on the cardinality of topological spaces, Soviet Math. Dokl.21.2 (1980), 506–509.[R] Ranchin D.V., On compactness modulo an ideal, Soviet Math. Dokl. 13.1 (1972), 193–197.Department of Mathematics, Ohio University, Athens, OH 45701, USA(Received January 23, 1996)

The sizes of relatively compact $T_1$-spaces

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