Polytopological Semantics for Intuitionistic Modal Logics

We develop polytopological semantics for various constructive, intuitionistic, and Gödel–Dummett variations of $\mathsf{K}\mathsf{4}$ and $\mathsf{S}\mathsf{4}$. In our models, intuitionistic and modal operators are interpreted via various topologies over a single set, equipped with either the closure or derivative operators. We identify regularity conditions to ensure that spaces validate each of our target logics and prove that all the logics considered are sound and strongly complete with respect to their respective semantics.