Modal Logics of Some Hereditarily Irresolvable Spaces

Abstract

A topological space is hereditarily k-irresolvable if none of its subspaces can be partitioned into k dense subsets. We use this notion to provide a topological semantics for a sequence of modal logics whose n-th member K4\(\mathbb {C}_n\) is characterised by validity in transitive Kripke frames of circumference at most n. We show that under the interpretation of the modality \(\Diamond \) as the derived set (of limit points) operation, K4\(\mathbb {C}_n\) is characterised by validity in all spaces that are hereditarily \(n+1\)-irresolvable and have the T\(_D\) separation property. We also identify the extensions of K4\(\mathbb {C}_n\) that result when the class of spaces involved is restricted to those that are crowded, or densely discrete, or openly irresolvable, the latter meaning that every non-empty open subspace is 2-irresolvable. Finally, we give a topological semantics for K4M, where M is the McKinsey axiom.