Exercise 14, § 4

a) Let G be a topological group operating continuously on a topological space X, and let $x_0$ be a point of X such that $s.x_0 = x_0$ for all $s \in G$. For each neighbourhood V of $x_0$ and each quasi-compact subset K of G, show that the set $\bigcap_{s \in K} s.V$ is a neighbourhood of $x_0$.

b) Deduce from a) that if G is a locally compact group, V a neighbourhood of e, K a compact subset of G, then the set $\bigcap_{s \in K} sVs^{-1}$ is a neighbourhood of e.