Proposition 11

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

General Topology, chapter III, § 2. Subgroups, quotient groups, homomorphisms homogeneous spaces, product groups

If the equivalence relation $S$ on $X$ is open and compatible with $G$ , then $G$ operates continuously on $X/S$ .

Since the relation of equality on $G$ and the relation $S$ on $X$ are open, it is enough to show that the mapping $(s, x) \to s.\psi(x) = \psi(s.x)$ of $G \times X$ into $X/S$ is continuous (Chapter I, § 5, no. 3, Corollary to Proposition 8); but this follows from the continuity of $\psi$ and of the mapping

(s, x) \to s.x.

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

Cited by

Proposition 12, § 2
Proposition 21, § 2

Cites

Chapter I, § 5, no. 3, Corollary to Proposition 8