Modal logic with the difference modality of topological $T_0$-spaces

The aim of the paper is to study the topological modal logic of $T_0$ spaces, with the difference modality (for $T_n$, where $n\ge 1$ the corresponding logics were known). We consider propositional modal logic with two modal operators $\mathrm{\square }$ and $[\mathrm{\ne }]$. $\mathrm{\square }$ is interpreted as an interior operator and $[\mathrm{\ne }]$ corresponds to the inequality relation. We introduce the logic $S4D{T}_0$ and show that $S4D{T}_0$ is the logic of all $T_0$ spaces and has the finite model property.