Exercise 9, § 4

Let $G$ be a topological group operating continuously on two spaces $X$ and $X'$, and let $f : X \to X'$ be a continuous mapping compatible with the identity mapping of $G$ ($\S$ 2, no. 4). Let $\bar{f} : X/G \to X'/G$ be the continuous mapping induced by $f$ on the quotient spaces.

a) Show that if $f$ is open (resp. closed, proper, injective, surjective) then so is $\bar{f}$. (To prove that if $f$ is proper then $\bar{f}$ is proper, show that the inverse image of a point of $X'/G$ under $\bar{f}$ is quasi-compact.)

b) Suppose that $G$ acts properly and freely on both $X$ and $X'$. Show that if $\bar{f}$ is proper, then so is $f$ [observe that the restriction of $f$ to an orbit $G.x$ is a homeomorphism onto $G.f(x)$].