Graph topologies induced by edge lengths

Let $G$ be a graph each edge $e$ of which is given a length $\ell(e)$. This naturally induces a distance $d_\ell(x,y)$ between any two vertices $x,y$, and we let $\ell-TOP$ denote the completion of the corresponding metric space. It turns out that several well studied topologies on infinite graphs are special cases of $\ell-TOP$. Moreover, it seems that $\ell-TOP$ is the right setting for studying various problems. The aim of this paper is to introduce $\ell-TOP$, providing basic facts, motivating examples and open problems, and indicate possible applications.