Proof That the Plane With the Euclidean Metric is Locally Compact
The planewith the euclidean metric is locally compact.
A spaceis said to be locally compact if, for anyand any neighbourhoodof there is a compact setsuch that
is not compact since it is not bounded.
Letand letbe a neighbourhood ofThen their existssuch that
The setis a closed and bounded subset ofso is compact and
Hencewith the Euclidean metric is locally compact.