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

TL;DR

This paper develops a modal logic framework for $T_0$ topological spaces incorporating a difference modality, establishing the logic $S4DT_0$ as complete and possessing the finite model property.

Contribution

It introduces the logic $S4DT_0$ for $T_0$ spaces with difference modality and proves its completeness and finite model property.

Findings

$S4DT_0$ characterizes all $T_0$ spaces.

$S4DT_0$ has the finite model property.

The logic extends known results for higher $T_n$ spaces.

Abstract

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.