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2015 | 231 | 2 | 189-208
Tytu艂 artyku艂u

Seven characterizations of non-meager 饾柉-filters

Tre艣膰 / Zawarto艣膰
Warianty tytu艂u
J臋zyki publikacji
EN
Abstrakty
EN
We give several topological/combinatorial conditions that, for a filter on 蠅, are equivalent to being a non-meager 饾柉-filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager 饾柉-filter. Here, we identify a filter with a subspace of $2^{蠅}$ through characteristic functions. Along the way, we generalize to non-meager 饾柉-filters a result of Miller (1984) about 饾柉-points, and we employ and give a new proof of results of Marciszewski (1998). We also employ a theorem of Hern谩ndez-Guti茅rrez and Hru拧谩k (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 饾柉-subfilter" is independent of 饾柟饾枼饾枹 (more precisely, it is a consequence of 饾敳 < 饾敜 and its negation is a consequence of 鈰). It follows from results of Hru拧谩k and van Mill (2014) that, under 饾敳 < 饾敜, a filter has less than 饾敔 types of countable dense subsets if and only if it is a non-meager 饾柉-filter. In particular, under 饾敳 < 饾敜, there exists an ultrafilter with 饾敔 types of countable dense subsets. We also show that such an ultrafilter exists under 饾柆 饾枲(countable).
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Tw贸rcy
  • Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 53706, U.S.A.
  • Kurt G枚del Research Center, for Mathematical Logic, University of Vienna, W盲hringer Stra脽e 25, A-1090 Wien, Austria
  • Kurt G枚del Research Center, for Mathematical Logic, University of Vienna, W盲hringer Stra脽e 25, A-1090 Wien, Austria
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm231-2-5
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