  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.
