We determine when the topological spaces $|G|$ naturally associated
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. The proof uses 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. The proof uses
ultrafilters and a lemma relating ends to directions.
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