A discrete Hopf-Rinow-theorem

Abstract

We prove a version of the Hopf-Rinow-theorem with respect to path metrics on discrete spaces. The novel aspect is that we do not a priori assume local finiteness but isolate a local finiteness type condition, called essential local finiteness, that is indeed necessary. As a side product we identify the maximal weight, called the geodesic weight, which generates the path metric in the situation when the space is complete with respect to any of the equivalent notions of completeness proven in the Hopf-Rinow theorem. As an application, we characterize the graphs for which the resistance metric is a path metric induced by the graph structure.