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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\geq1 $ the correspondinglogics were known). We consider propositional modal logic with two modaloperators $\square$ and $[\ne]$. $\square$ is interpreted as an interioroperator and $[\ne]$ corresponds to the inequality relation. We introduce thelogic $S4DT_0$ and show that $S4DT_0$ is the logic of all $T_0$ spaces and hasthe finite model property.

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