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**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 \(p\in X\).

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 |