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 ${\bf Baire}(X)$. 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.

Theorems & Definitions (67)

• Definition 2.1
• Remark 2.2
• Theorem 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Theorem 2.8
• proof : Proof sketch
• Theorem 2.9: Scroggs' Theorem