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arXiv · 1508.00124

Non-meager free sets and independent families

Andrea Medini·Dušan Repovš·Lyubomyr Zdomskyy
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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 $|\mathcal{R}|\leqω$, 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 $ω$, and define the corresponding cardinal invariant. Furthermore, assuming Martin's Axiom for countable posets, our result can be strengthened by substituting "$|\mathcal{R}|\leqω$" with "$|\mathcal{R}|<\mathfrak{c}$" and "Baire" with "completely Baire".

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Andrea Medini, Dušan Repovš, Lyubomyr Zdomskyy. 2017-06-20. Non-meager free sets and independent families. https://doi.org/10.1090/proc%2F13513

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