Corollary 1
Tag 0BKV, permanent. Cite it as [Bourbaki, Tag 0BKV].
General Topology, chapter III, § 4. Groups operating properly on a topological space; compact ness in topological groups and spaces with operators
If A is a closed (resp. compact) subset of X, then K.A is closed in X (resp. compact if X is Hausdorff).
The assertion concerning closed sets follows from Proposition 1 and the fact that a proper mapping is closed (Chapter I, § 10, no. 1, Proposition 1). The assertion concerning compact sets is trivial.
It should be noted that if L is a compact subset of X, and F a closed subset of G, then F.L is not necessarily closed in X (, Exercise 29; cf. no. 5, Corollary to Proposition 12).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
Cited by
- Lemma 3, § 9
- Example 3, § 1
- Exercise 13, § 4
- Exercise 19, § 4
- Corollary 2, § 4
- Proposition 2, § 4
- Exercise 18, § 6
- Lemma 2, § 7
- Proposition 3, § 7