Non-meager free sets and independent families

TL;DR

This paper proves the existence of dense, nowhere meager subsets in Polish spaces that are free of meager relations, and applies this to construct non-meager independent families on , introducing related cardinal invariants.

Contribution

It generalizes previous results by establishing dense free sets for countable collections of meager relations on Polish spaces and explores their applications to independent families.

Findings

Existence of dense Baire subspaces free of meager relations

Construction of non-meager independent families on 

Introduction of a new cardinal invariant related to independent families

Abstract

Our main result is that, given a collection $\mathcal{R}$ of meager relations on a Polish space $X$ such that $|\mathcal{R}|\leq\omega$, 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 "$|\mathcal{R}|\leq\omega$" with "$|\mathcal{R}|<\mathfrak{c}$" and "Baire" with "completely Baire".