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 $\mathrm{non}(\mathcal M_X) \leq \mathrm{cf}[\kappa]^\omega \cdot \mathrm{non}(\mathcal M_{\mathbb R})$. 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.