Analytic countably splitting families. (English) Zbl 1067.03054

Spinas, Otmar

J. Symb. Log. 69, No. 1, 101-117 (2004).

Summary: A family \(A\subseteq{\mathcal P}(\omega)\) is called countably splitting if, for every countable \(F\subseteq [\omega]^\omega\), some element of \(A\) splits every member of \(F\). We define a notion of a splitting tree by means of which we prove that every analytic countably splitting family contains a closed countably splitting family. An application of this notion solves a problem of Blass. On the other hand we show that there exists an \(F_\sigma\) splitting family that does not contain a closed splitting family.

MSC:

03E05	Other combinatorial set theory
03E15	Descriptive set theory
03E17	Cardinal characteristics of the continuum
54A35	Consistency and independence results in general topology

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