Graph topologies induced by edge lengths

Let GG be a graph each edge ee of which is given a length ℓ(e)ℓ(e). This naturally induces a distance dℓ(x,y)dℓ(x,y) between any two vertices x,yx,y, and we let |G|ℓ|G|ℓ denote the completion of the corresponding metric space. It turns out that several well-studied topologies on infinite graphs are special cases of |G|ℓ|G|ℓ. Moreover, it seems that |G|ℓ|G|ℓ is the right setting for studying various problems. The aim of this paper is to introduce |G|ℓ|G|ℓ, providing basic facts, motivating examples and open problems, and indicate possible applications.Parts of this work suggest interactions between graph theory and other fields, including algebraic topology and geometric group theory.