Proposition 10

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

General Topology, chapter III, § 4. Groups operating properly on a topological space; compact ness in topological groups and spaces with operators

Let $G$ be a topological group operating continuously on a locally compact space $X$ , and suppose that $X/G$ is Hausdorff. Let $\varphi$ be the canonical mapping of $X$ onto $X/G$ . Then if $K'$ is any compact subset of $X/G$ , there is a compact subset $K$ of $X$ such that $\varphi(K) = K'$ .

Since the equivalence relation defined by $G$ is open ( $§ 2$ , no. 4, Lemma 2), the proposition is a particular case of Proposition 10 of Chapter I, $§ 10$ , no. 4.

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

Cited by

Exercise 13, § 4
Exercise 15, § 4
Proposition 9, § 4

Cites

Nothing.