Proof That the Plane With the Euclidean Metric is Locally Compact

Theorem

The planewith the euclidean metric is locally compact.

Proof

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.