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 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 Baire(X ), and that soundness and strong completeness also holds in the language with the universal modality.