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09 February 2025
 
  » arxiv » 1508.0124

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Non-meager free sets and independent families
Andrea Medini ; Dušan Repovš ; Lyubomyr Zdomskyy ;
Date 1 Aug 2015
AbstractOur main result is that, given a collection $mathcal{R}$ of meager relations on a Polish space $X$ such that $|mathcal{R}|leqomega$, there exists a dense Baire subspace $F$ of $X$ (equivalently, a nowhere meager subset of $X$) such that $F$ is $R$-free for every $Rinmathcal{R}$. This generalizes a recent result of Banakh and Zdomskyy. Furthermore, assuming Martin’s Axiom for countable posets, our result can be strengthened by substituting "$|mathcal{R}|leqomega$" with "$|mathcal{R}|<mathfrak{c}$" and "Baire" with "completely Baire". As an application, we show that there exists a non-meager independent family on $omega$, and define the corresponding cardinal invariant.
Source arXiv, 1508.0124
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