The Baire closure and its logic

Abstract

The Baire algebra of a topological space $X$ is the quotient of the algebra of all subsets of $X$ modulo the meager sets. We show that this Boolean algebra can be endowed with a natural closure operator, resulting in a closure algebra which we denote $\mathbf{Baire}(X)$. We identify the modal logic of such algebras to be the well-known system $S5$, and prove soundness and strong completeness for the cases where $X$ is crowded and either completely metrizable and continuum-sized or locally compact Hausdorff. We also show that every extension of $S5$ is the modal logic of a subalgebra of $\mathbf{Baire}(X)$, and that soundness and strong completeness also holds in the language with the universal modality.