On Modal Logics of Connectedness in Metric Spaces

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.