End spaces and spanning trees

We determine when the topological spaces |G| naturally associated with a graph G and its ends are metrizable or compact. In the most natural topology, |G| is metrizable if and only if G has a normal spanning tree. We give two proofs, one of them based on Stone’s theorem that metric spaces are paracompact. We show that |G| is compact in the most natural topology if and only if no finite vertex separator of G leaves infinitely many components. When G is countable and connected, this is equivalent to the existence of a locally finite spanning tree. The proof uses ultrafilters and a lemma relating ends to directions.