Proposition 9

Tag 0B8N, permanent. Cite it as [Bourbaki, Tag 0B8N].

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

Every locally compact space is regular.

Let $X$ be a locally compact space, then every point $x \in X$ has a compact neighbourhood $V$ ; since $X$ is Hausdorff, $V$ is closed (no. 3, Proposition 4). On the other hand, $V$ is a regular subspace of $X$ (no. 2, Corollary to Proposition 1) and therefore $X$ is regular (§ 8, no. 4, Proposition 13).

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

Cited by

Nothing.

Cites

§ 8, no. 4, Proposition 13
Corollary to Proposition 1
no. 3, Proposition 4