Let MX 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 non(MX), is exactly non(MX)=cf[κ]ω⋅non(MR), where κ is the minimum weight of a nonempty open subset of X . We also characterize the additivity and covering numbers for MX in terms of simple topological properties of X . Some bounds are proved and some questions raised concerning the cofinality of MX 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 π -weight κ , then non(MX)≤cf[κ]ω⋅non(MR). 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. © 2025 The Authors.
