####

#### 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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