# Strong completeness of modal logics over 0-dimensional metric spaces

## TL;DR

This paper establishes strong completeness results for certain modal logics with the universal modality over 0-dimensional metric spaces, and demonstrates limitations of standard systems due to compactness failure.

## Contribution

It proves strong completeness for modal logics over 0-dimensional dense metric spaces and shows non-strong completeness in some cases due to compactness issues.

## Key findings

- Strong completeness for modal logics over 0-dimensional spaces
- Failure of compactness prevents strong completeness in some languages
- Identification of conditions for strong completeness in topological semantics

## Abstract

We prove strong completeness results for some modal logics with the universal modality, with respect to their topological semantics over 0-dimensional dense-in-themselves metric spaces. We also use failure of compactness to show that, for some languages and spaces, no standard modal deductive system is strongly complete.