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Non-meager free sets and independent families


Authors: Andrea Medini, Dušan Repovš and Lyubomyr Zdomskyy
Journal: Proc. Amer. Math. Soc. 145 (2017), 4061-4073
MSC (2010): Primary 54E50, 54E52; Secondary 03E05, 03E50
DOI: https://doi.org/10.1090/proc/13513
Published electronically: April 4, 2017
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Abstract: Our main result is that, given a collection $ \mathcal {R}$ of meager relations on a Polish space $ X$ such that $ \vert\mathcal {R}\vert\leq \omega $, there exists a dense Baire subspace $ F$ of $ X$ (equivalently, a nowhere meager subset $ F$ of $ X$) such that $ F$ is $ R$-free for every $ R\in \mathcal {R}$. This generalizes a recent result of Banakh and Zdomskyy. As an application, we show that there exists a non-meager independent family on $ \omega $, and define the corresponding cardinal invariant. Furthermore, assuming Martin's Axiom for countable posets, our result can be strengthened by substituting `` $ \vert\mathcal {R}\vert\leq \omega $'' with `` $ \vert\mathcal {R}\vert<\mathfrak{c}$'' and ``Baire'' with ``completely Baire''.


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Additional Information

Andrea Medini
Affiliation: Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Währinger Straße 25, A-1090 Wien, Austria
Email: andrea.medini@univie.ac.at

Dušan Repovš
Affiliation: Faculty of Education, and Faculty of Mathematics and Physics, University of Ljubljana, Kardeljeva Ploščad 16, Ljubljana, 1000, Slovenia
Email: dusan.repovs@guest.arnes.si

Lyubomyr Zdomskyy
Affiliation: Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Währinger Straße 25, A-1090 Wien, Austria
Email: lyubomyr.zdomskyy@univie.ac.at

DOI: https://doi.org/10.1090/proc/13513
Keywords: Free set, meager relation, completely Baire, hereditarily Baire, independent family.
Received by editor(s): August 1, 2015
Received by editor(s) in revised form: September 30, 2016
Published electronically: April 4, 2017
Additional Notes: The first-listed author acknowledges the support of the FWF grant M 1851-N35
The second-listed author acknowledges the support of the SRA grant P1-0292-0101
The third-listed author acknowledges the support of the FWF grant I 1209-N25. The third-listed author also thanks the Austrian Academy of Sciences for its generous support through the APART Program.
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2017 American Mathematical Society

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