Proposition 2
Tag 0BKY, permanent. Cite it as [Bourbaki, Tag 0BKY].
General Topology, chapter III, § 4. Groups operating properly on a topological space; compact ness in topological groups and spaces with operators
Let K be a compact group operating continuously on a Hausdorff space X. Then:
a) K operates properly on X. b) The mapping (s, x) → s.x of K × X into X is proper. c) The canonical mapping of X onto X/K is proper.
b) is a consequence of Proposition 1. As to a), since K is compact, pr₂ : (s, x) → x is proper (Chapter I, § 10, no. 2, Theorem 1, Corollary 5); hence, X being Hausdorff, (s, x) → (x, s.x) is proper (Chapter I, § 10, no. 1, Proposition 5, Corollary 3). It remains to prove c). By Corollary 1 to Proposition 1, the canonical mapping φ : X → X/K is closed. If Z is any topological space and if we make K operate trivially on Z, then K operates continuously on X × Z and hence the canonical mapping X × Z → (X × Z)/K is closed. But (X × Z)/K can be canonically identified with (X/K) × Z (§ 2, no. 4, Lemma 2 and Chapter I, § 5, no. 3, Corollary to Proposition 8); hence the canonical mapping X × Z → (X × Z)/K can be identified with φ × 1, and since it is closed for all Z it follows that φ is proper.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.