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. 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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