## A weakening of $$\clubsuit$$, with applications to topology.(English)Zbl 0676.54005

With $$L_ 1$$ the set of countable limit ordinals, the author denotes by (t) the following combinatorial principle: there are an $$\omega_ 1$$-sequence $$\{S_{\lambda}:$$ $$\lambda \in L_ 1\}$$, with each $$S_{\lambda}$$ a cofinal $$\omega$$-sequence in $$\lambda$$, and for each $$\lambda$$ a disjoint partition $$S_{\lambda}=\cup_{n<\omega}S_{\lambda}(n)$$, such that if $$X\subseteq \omega_ 1$$ with $$| X| =\omega_ 1$$ then for some $$\lambda \in L_ 1$$ (depending on X) we have $$| X\cap S_{\lambda}(n)| =\omega$$ 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.
The author shows (Theorem 2.1) that (t) implies the existence of an Ostaszewski topology for $$\omega_ 1$$-that is, a locally compact Hausdorff topology $${\mathcal J}$$ in which each initial segment in $$\omega_ 1$$ is $${\mathcal J}$$-open and each $${\mathcal J}$$-open set is either countable or co-countable; from (t)$$+CH$$ one may arrange further that $$<\omega_ 1,{\mathcal J}>$$ 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 $$\chi (X)=\omega_ 1$$ for each $$p\in X$$; such a space X cannot be constructed in ZFC, since 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 $$p\in X$$.
Reviewer: W.W.Comfort

### MSC:

 54A35 Consistency and independence results in general topology 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 54D30 Compactness
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