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Zbl 0676.54005
Juhász, István
A weakening of $\clubsuit$, with applications to topology.
(English)
[J] Commentat. Math. Univ. Carol. 29, No.4, 767-773 (1988). ISSN 0010-2628

With $L\sb 1$ the set of countable limit ordinals, the author denotes by (t) the following combinatorial principle: there are an $\omega\sb 1$-sequence $\{S\sb{\lambda}:$ $\lambda \in L\sb 1\}$, with each $S\sb{\lambda}$ a cofinal $\omega$-sequence in $\lambda$, and for each $\lambda$ a disjoint partition $S\sb{\lambda}=\cup\sb{n<\omega}S\sb{\lambda}(n)$, such that if $X\subseteq \omega\sb 1$ with $\vert X\vert =\omega\sb 1$ then for some $\lambda \in L\sb 1$ (depending on X) we have $\vert X\cap S\sb{\lambda}(n)\vert =\omega$ for all n. That $\clubsuit$ implies (t) is clear; that (t) is consistently weaker than $\clubsuit$ is given by Theorem 1.2: If a single Cohen real is added to a model of ZFC, then (t) holds in the resulting extension. \par The author shows (Theorem 2.1) that (t) implies the existence of an Ostaszewski topology for $\omega\sb 1$-that is, a locally compact Hausdorff topology ${\cal J}$ in which each initial segment in $\omega\sb 1$ is ${\cal J}$-open and each ${\cal J}$-open set is either countable or co-countable; from (t)$+CH$ one may arrange further that $<\omega\sb 1,{\cal J}>$ is countably compact. In a much more complicated inverse limit construction, the author shows that (t) implies the existence of a compact Hausdorff space X such that $t(X)=\pi(X)=\omega$, but $\chi (X)=\omega\sb 1$ for each $p\in X$; such a space X cannot be constructed in ZFC, since {\it A. Dow} has shown [Topology Proc. 13, No.1, 17-72 (1988)] from PFA that if X is compact and $t(X)=\omega$, then $\chi (p,X)=\omega$ for some $p\in X$.
[W.W.Comfort]
MSC 2000:
*54A35 Consistency and independence results (general topology)
54A25 Cardinality properties of topological spaces
54D30 Compactness

Keywords: cardinal functions; Ostaszewski space; tightness; combinatorial principle; Cohen real; model of ZFC; (t)+CH

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