Exercise 15, § 4

Let G be a locally compact group and let $G_0$ be the identity component in G; suppose that the quotient group $G/G_0$ is compact.

a) Show that G is $\sigma$-compact (remark that $G_0$ is $\sigma$-compact and use Proposition 10).

b) Let $(U_n)$ be a decreasing sequence of neighbourhoods of e in G. Show that there is a compact normal subgroup K of G, contained in the intersection of the $U_n$, and such that the identity element of $G/K$ has a countable fundamental system of neighbourhoods. [There exists a compact symmetric neighbourhood $V_0$ of e in G such that $G = V_0G_0$. Define recursively a sequence of compact symmetric neighbourhoods $V_n$ of e such that $V_n^2 \subset V_{n-1} \cap U_n$ and $xV_nx^{-1} \subset V_{n-1}$ for each $x \in V_0$ (Exercise 14 b); show that the intersection K of the $V_n$ answers the question].

c) Let X be a locally compact space which has a countable base; suppose G operates continuously on X in such a way that there is no $s \neq e$ in G for which $s.x = x$ for all $x \in X$. Show that the identity element of G has a countable fundamental system of neighbourhoods. [Let $(W_n)$ be a countable base of the topology of X, consisting of relatively compact sets; for each pair of integers $(m, n)$ such that $W_m \subset W_n$, let $U_{mn}$ be the set of all $s \in G$ such that $s.W_m \subset W_n$; show that the $U_{mn}$ are open in G and apply the result of b).]