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On the combinatorial principle P(\({\mathfrak c})\). (English) Zbl 0581.03038
The author gives a quite surprising result (Theorem 1.2): P(\({\mathfrak c})\) implies Martin’s axiom for \(\sigma\)-centered posets. This establishes the equivalence of these two statements. The combinatorial statement P(\({\mathfrak c})\) is as follows. For each centered collection \({\mathcal A}\) of fewer than \({\mathfrak c}\) subsets of \(\omega\), there is an infinite \(B\subseteq \omega\) such that \(B\setminus A\) is finite for all \(A\in {\mathcal A}\). Martin’s axiom for \(\sigma\)-centered posets says that, for any \(\sigma\)-centered poset P (i.e. union of countably many centered subsets) and a collection \({\mathcal D}\) of fewer than \({\mathfrak c}\) dense subsets of P, there is a \({\mathcal D}\)-generic \(G\subseteq P\). The proof is combinatorial, and it makes full use of a consequence of P(\({\mathfrak c})\), i.e. BB(\({\mathfrak c}):\) For each collection \({\mathcal K}\) of fewer than \({\mathfrak c}\) mappings from \(\cup_{n<\omega}^ n\omega\) to \(\omega\), there is an f:\(\omega\) \(\to \omega\) such that for each \(h\in {\mathcal K}\) there is an \(N_ h<\omega\) such that h(f\(| n)\leq f(n)\) for each \(n\geq N_ h\). To the reviewer’s knowledge, the above result has another shorter and more topological proof (offered by C. Mills).
Another main result of the present paper is the following: P(\({\mathfrak c})\) implies that there is a first countable Dowker space, where a Dowker space is a normal non-countably paracompact space. M. E. Rudin [ibid. 73, 179-186 (1971; Zbl 0224.54019)] first gave a real Dowker space, unfortunately with large cardinal functions, and asked if there exist Dowker spaces with small cardinal functions. In particular, whether there is a real first countable Dowker space remains open.

MSC:
03E05 Other combinatorial set theory
03E50 Continuum hypothesis and Martin’s axiom
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
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