Modal logics of Stone spaces (Q438813)

A Stone space is an image of a Boolean algebra under Stone duality. Using their previous results, the authors characterize modal propositional logics of Stone spaces. For a metrizable Stone space \(X\) its logic is S4 if \(X\) is not weakly scattered, and is S4 or S4.\({\mathrm{Grz}_n}\) for some \(n>1\) if \(X\) is scattered. Similar results are obtained for intermediate logics.