Baire theorem for ideals of sets

TL;DR

This paper investigates a Baire-type property for ideals on natural numbers, characterizing those with this property among classes like analytic and P-ideals, and explores related covering properties involving compact and meager sets.

Contribution

It provides characterizations of ideals satisfying the Baire-type property and identifies which ideals among certain classes possess this property.

Findings

Identifies ideals with the Baire property among analytic and P-ideals.

Provides multiple characterizations of these ideals.

Discusses related covering properties for compact and meager sets.

Abstract

We study ideals $\mathcal{I}$ on $\mathbb{N}$ satisfying the following Baire-type property: if $X$ is a complete metric space and $\{X_{A} \colon A \in \mathcal{I} \}$ is a family of nowhere dense subsets of $X$ with $X_{A} \subset X_{B}$ whenever $A \subset B$, then $\bigcup_{A \in \mathcal{I}}{X_{A}} \neq X$. 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 $X$.