Proposition 4

Let $G$ be a topological group operating properly on a topological space $X$, and let $x$ be a point of $X$. Let $G.x$ denote the orbit of $x$, and let $K_x$ denote the stabilizer of $x$. Then:

a) The mapping $s \to s.x$ of $G$ into $X$ is proper.

b) $K_x$ is quasi-compact.

c) The canonical mapping of $G/K_x$ onto $G.x$ is a homeomorphism.

d) The orbit $G.x$ is closed in $X$.

The inverse image of $\{ x \} \times X$ under $\theta : (s, x) \to (x, s.x)$ is $G \times \{ x \}$; hence, by Proposition 3 of Chapter I, § 10, no. 1, the restriction of $\theta$ to $G \times \{ x \}$ is a proper mapping of $G \times \{ x \}$ into $\{ x \} \times X$, whence a) follows. Since $K_x$ is the inverse image of $x$ under $s \to s.x$, b) follows from Chapter I, § 10, no. 2, Theorem 1. c) and d) are consequences of a), by virtue of Chapter I, § 10, no. 1, Propositions 2 and 5 b).