Baire theorem for ideals of sets

Abstract

We study ideals $\mathcal{I}$ on $\mathbb{N}$ satisfying the following Baire-type property: if $X$ is a complete metric space and $\{X_A:A\in \mathcal{I}\}$ is a family of nowhere dense subsets of $X$ with $X_A\subset X_B$ whenever $A\subset B$, then ${\bigcup }_{A\in \mathcal{I}}{X}_A\ne X$. We give several characterizations and determine the ideals having this property among certain classes like analytic ideals and P-ideals. We also discuss similar covering properties when considering families of compact and meager subsets of $X$.

To appear in Journal of Mathematical Analysis and Applications