Proposition 12

Tag 0B8R, permanent. Cite it as [Bourbaki, Tag 0B8R].

General Topology, chapter I, § 9. Compact spaces and locally compact spaces

In a Hausdorff space X, every locally compact subspace A is locally closed.

By hypothesis, for every $x \in A$ there is a neighbourhood V of x in X such that $V \cap A$ is compact and therefore closed in V (no. 3, Proposition 4).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Corollary, § 3

Cites

no. 3, Proposition 4