Example 1

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

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

Let H be a closed subgroup of a topological group G. If G operates properly on X, then so does H, since $H \times X$ is closed in $G \times X$ (Chapter I, § 10, no. 1, Proposition 5, Corollary 1). If for example we take $X = G$ , with G operating on itself by left translations, then the mapping $G \times X \to X \times X$ is a homeomorphism, hence proper; thus H operates properly on G by left translations.

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

Cited by

Nothing.

Cites

Chapter I, § 10, no. 1, Proposition 5
Corollary 1