Exercise 10, § 4

Let $G$ be a topological group and let $X, Y$ be two topological spaces on which $G$ operates continuously. Then $G$ operates continuously on $X \times Y$ according to the law $(s, (x, y)) \to (s.x, s.y)$. Let $(X \times Y)/G$ be denoted by $X \times ^G Y$.

a) Show that $X \times ^G G$ ($G$ operating by left translations on itself) is canonically homeomorphic to $X$. (Remark that if $U$ is open in $X$, then the union of the orbits of the points of $U \times \{e\}$ is open in $X \times G$).

b) Let $Y'$ be a third topological space on which $G$ operates continuously, and let $f : Y \to Y'$ be a continuous mapping compatible with the identity mapping of $G$. The mapping $i_X \times f$ induces a continuous mapping $\overline{f} : X \times ^G Y \to X \times ^G Y'$. If $f$ is open (resp. proper, injective, surjective) then so is $\overline{f}$ (Exercise 9). Give an example in which $f$ is closed, $G$ operates properly on $X$, but $\overline{f}$ is not closed [take $X = G$, $X' = P$ (a space consisting of a single point) and use Exercise 4].

c) Show that if $G$ operates properly on $X$, then $G$ operates properly on $X \times Y$.

d) Show that, if $Y$ is compact, the canonical mapping $X \times ^G Y \to X/G$ is proper [use b)].

e) Suppose that $G$ is locally compact, and let $G'$ be the one-point compactification of $G$. If $\omega$ is the point at infinity in $G'$, then $G$ operates continuously on $G'$ according to the external law given by $s.t = st$ if $t \in G$, and $s.\omega = \omega$; $X$ is then homeomorphic to an open subspace of $X \times ^G G'$ (use b)). Deduce that if $G$ operates properly on $X$ and if in addition $X/G$ is locally compact, then $X$ is locally compact [use c) to show that $X \times ^G G'$ is Hausdorff].

f) Give an example of a locally compact group $G$ operating continuously on a space $X$ which is not locally compact, such that $X/G$ consists of a single point (cf. § 2, Exercise 29). \