On Modal Logics of Connectedness in Metric Spaces

2026-06-30

AI Summary

This study addresses distance-based a-connectivity in metric spaces and topological connectivity by constructing corresponding modal logical systems. By introducing distance modal operators, topological modalities, and a universal modality, the work provides the first complete axiomatizations for both notions of connectedness and establishes that the resulting logics enjoy the finite model property. This contribution not only lays a formal logical foundation for reasoning about connectedness but also significantly extends the expressive and inferential capabilities of modal logic within metric and topological structures.

Abstract

For a positive number a, each metric space carries the relation D_a consisting of those pairs that are of distance less than a apart. A space X is said to be a-connected, if the graph (X,D_a) is connected (that is, there is a D_a-path between every pair of points in X). We give a complete axiomatization of a-connected metric spaces in the language with a family of distance modalities and the universal modality. Then we give a complete axiomatization of the logic of connected (in the classical topological sense) metric spaces in the language with the topological modality, universal modality, and a single distance modality. We also show that these logics have the finite model property.

Problem

• modal logic
• connectedness
• metric spaces
• distance modality
• topological modality

Innovation

• modal logic
• metric spaces
• connectedness
• axiomatization
• finite model property