The Baire closure and its logic

TL;DR

This paper introduces a closure operator on the Baire algebra of a topological space, linking it to the modal logic S5, and proves completeness results for certain classes of spaces.

Contribution

It defines a natural closure operator on the Baire algebra and establishes its modal logic as S5, with completeness proofs for specific topological spaces.

Findings

The modal logic of the Baire algebra with closure is S5.

Soundness and completeness are proved for crowded, metrizable, continuum-sized spaces.

Every extension of S5 corresponds to a subalgebra of the Baire algebra.

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 ${\bf Baire}(X)$. We identify the modal logic of such algebras to be the well-known system $\sf S5$, 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 $\sf S5$ is the modal logic of a subalgebra of ${\bf Baire}(X)$, and that soundness and strong completeness also holds in the language with the universal modality.