Topological Modal Logics with Difference Modality

We consider propositional modal logic with two modal operators $\mathrm{\square }$ and $\text{D}$. In topological semantics $\mathrm{\square }$ is interpreted as an interior operator and $\text{D}$ as difference. We show that some important topological properties are expressible in this language. In addition, we present a few logics and proofs of f.m.p. and of completeness theorems.