Seven characterizations of non-meager 𝖯-filters

• Autorzy: Kenneth Kunen , Andrea Medini , Lyubomyr Zdomskyy
• Języki publikacji: EN

Abstrakty

EN

We give several topological/combinatorial conditions that, for a filter on ω, are equivalent to being a non-meager 𝖯-filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager 𝖯-filter. Here, we identify a filter with a subspace of $2^{ω}$ through characteristic functions. Along the way, we generalize to non-meager 𝖯-filters a result of Miller (1984) about 𝖯-points, and we employ and give a new proof of results of Marciszewski (1998). We also employ a theorem of Hernández-Gutiérrez and Hrušák (2013), and answer two questions that they posed. Our result also resolves several issues raised by Medini and Milovich (2012), and proves false one "theorem" of theirs. Furthermore, we show that the statement "Every non-meager filter contains a non-meager 𝖯-subfilter" is independent of 𝖹𝖥𝖢 (more precisely, it is a consequence of 𝔲 < 𝔤 and its negation is a consequence of ⋄). It follows from results of Hrušák and van Mill (2014) that, under 𝔲 < 𝔤, a filter has less than 𝔠 types of countable dense subsets if and only if it is a non-meager 𝖯-filter. In particular, under 𝔲 < 𝔤, there exists an ultrafilter with 𝔠 types of countable dense subsets. We also show that such an ultrafilter exists under 𝖬 𝖠(countable).

Kategorie tematyczne

• 03E05: Other combinatorial set theory
• 54G99: None of the above, but in this section
• 03E35: Consistency and independence results

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2015
• Tom: 231
• Numer: 2
• Strony: 189-208

Daty

wydano

2015

Twórcy

autor

Kenneth Kunen

• Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 53706, U.S.A.

autor

Andrea Medini

• Kurt Gödel Research Center, for Mathematical Logic, University of Vienna, Währinger Straße 25, A-1090 Wien, Austria

autor

Lyubomyr Zdomskyy

• Kurt Gödel Research Center, for Mathematical Logic, University of Vienna, Währinger Straße 25, A-1090 Wien, Austria

Identyfikatory

DOI

10.4064/fm231-2-5