2021/02/06 by Guram Bezhanishvili, Bezhanishvili, Guram, David Fernández–Duque +1
Computer Science · #Logic, Reasoning, and Knowledge
paper · pdf · doi:10.48550/arxiv.2102.03564
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.