Seven characterizations of non-meager 𝖯-filters

Kenneth Kunen, Andrea Medini, and Lyubomyr Zdomskyy. "Seven characterizations of non-meager 𝖯-filters." Fundamenta Mathematicae 231.2 (2015): 189-208. <http://eudml.org/doc/286570>.

@article{KennethKunen2015,
abstract = {We give several topological/combinatorial conditions that, for a filter on ω, are equivalent to being a non-meager -filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager -filter. Here, we identify a filter with a subspace of $2^\{ω\}$ through characteristic functions. Along the way, we generalize to non-meager -filters a result of Miller (1984) about -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 -subfilter” is independent of (more precisely, it is a consequence of < and its negation is a consequence of ⋄). It follows from results of Hrušák and van Mill (2014) that, under < , a filter has less than types of countable dense subsets if and only if it is a non-meager -filter. In particular, under < , there exists an ultrafilter with types of countable dense subsets. We also show that such an ultrafilter exists under (countable).},
author = {Kenneth Kunen, Andrea Medini, Lyubomyr Zdomskyy},
journal = {Fundamenta Mathematicae},
keywords = {filter; ultrafilter; P-filter; countable dense homogeneity; completely Baire},
language = {eng},
number = {2},
pages = {189-208},
title = {Seven characterizations of non-meager 𝖯-filters},
url = {http://eudml.org/doc/286570},
volume = {231},
year = {2015},
}

TY - JOUR
AU - Kenneth Kunen
AU - Andrea Medini
AU - Lyubomyr Zdomskyy
TI - Seven characterizations of non-meager 𝖯-filters
JO - Fundamenta Mathematicae
PY - 2015
VL - 231
IS - 2
SP - 189
EP - 208
AB - We give several topological/combinatorial conditions that, for a filter on ω, are equivalent to being a non-meager -filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager -filter. Here, we identify a filter with a subspace of $2^{ω}$ through characteristic functions. Along the way, we generalize to non-meager -filters a result of Miller (1984) about -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 -subfilter” is independent of (more precisely, it is a consequence of < and its negation is a consequence of ⋄). It follows from results of Hrušák and van Mill (2014) that, under < , a filter has less than types of countable dense subsets if and only if it is a non-meager -filter. In particular, under < , there exists an ultrafilter with types of countable dense subsets. We also show that such an ultrafilter exists under (countable).
LA - eng
KW - filter; ultrafilter; P-filter; countable dense homogeneity; completely Baire
UR - http://eudml.org/doc/286570
ER -