Derivational modal logics with the difference modality

TL;DR

This paper explores modal logics combining derivational and difference modalities over various topological spaces, providing axiomatizations and completeness proofs for multiple classes, and analyzing language expressiveness.

Contribution

It introduces axiomatizations and completeness results for derivational and difference modal logics across diverse topological spaces, expanding understanding of their definability and relationships.

Findings

Axiomatizations for all spaces and specific subclasses.

Completeness proofs for multiple classes of topological spaces.

Analysis of language definability with different modality combinations.

Abstract

In this chapter we study modal logics of topological spaces in the combined language with the derivational modality and the difference modality. We give axiomatizations and prove completeness for the following classes: all spaces, $T_1$-spaces, dense-in-themselves spaces, a zero-dimensional dense-in-itself separable metric space, $R^n$ $(n\ge 2)$. We also discuss the correlation between languages with different combinations of the topological, the derivational, the universal and the difference modality in terms of definability.