Cardinal invariants of a meager ideal

Abstract

Let ${\mathcal{M}}_X$ denote the ideal of meager subsets of a topological space $X$. We prove that if $X$ is a completely metrizable space without isolated points, then the smallest cardinality of a non-meager subset of $X$, denoted $\mathrm{non}({\mathcal{M}}_X)$, is exactly $\mathrm{non}({\mathcal{M}}_X)=\mathrm{cf}[\kappa {]}^{\omega }\cdot \mathrm{non}({\mathcal{M}}_{\mathbb{R}})$, where $\kappa$ is the minimum weight of a nonempty open subset of $X$. We also characterize the additivity and covering numbers for ${\mathcal{M}}_X$ in terms of simple topological properties of $X$. Some bounds are proved and some questions raised concerning the cofinality of ${\mathcal{M}}_X$ and the cofinality of the related ideal of nowhere dense subsets of $X$. We also show that if $X$ is a compact Hausdorff space with $\pi$-weight $\kappa$, then $\mathrm{non}({\mathcal{M}}_X)\le \mathrm{cf}[\kappa {]}^{\omega }\cdot \mathrm{non}({\mathcal{M}}_{\mathbb{R}})$. This bound for compact Hausdorff spaces is not sharp, in the sense that it is consistent for such a space to have non-meager subsets of even smaller cardinality.