Exercise 18, § 9

Let G be a Lie group operating properly on a completely regular topological space E. a) Let $x\in E$, and let $t$ be its orbit type; show that there exists an open neighbourhood U of $x$, stable under G, such that for all $u\in U$ the type of $u$ is $\geq t$.

b) Assume that G operates freely on E; let $\pi : E\rightarrow E/G$ be the canonical projection. Show that, for every point $z$ of $E/G$, there exists an open neighbourhood U of $z$ and a continuous map $s: U\rightarrow E$ such that $\pi \circ s(u) =u$ for all $u\in U$.