  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 corresponding
logics were known). We consider propositional modal logic with two modal
operators $\square$ and $[\ne]$. $\square$ is interpreted as an interior
operator and $[\ne]$ corresponds to the inequality relation. We introduce the
logic $S4DT_0$ and show that $S4DT_0$ is the logic of all $T_0$ spaces and has
the finite model property.
