Analytic countably splitting families. (English) Zbl 1067.03054

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.


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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[1] Annals of Mathematics Studies 52 pp 85– (1964)
[2] DOI: 10.1016/0168-0072(94)00027-Z · Zbl 0824.03025
[3] Handbook of Set Theory
[4] Set theory (Annual Boise extravaganza in set theory conference, 1992–94) 192 pp 31– (1996)
[5] Set theory of the reals 6 pp 619– (1993)
[6] DOI: 10.1016/0168-0072(94)90025-6 · Zbl 0821.03021
[7] DOI: 10.1090/S0002-9947-1977-0450070-1
[8] DOI: 10.1002/1521-3870(200211)48:4<517::AID-MALQ517>3.0.CO;2-# · Zbl 1017.03026
[9] DOI: 10.2307/1971035 · Zbl 0336.02049
[10] Non-constructive Galois-Tukey connections 62 pp 1179– (1997)
[11] Topology 1 (1980) · Zbl 0459.01022
[12] Set theory, An introduction to independence proofs (1983) · Zbl 0534.03026
[13] Classical descriptive set theory (1995)
[14] Comptes Rendus de l’Académie des Sciences Paris, Série A 281 pp 85– (1975)
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