Exercise 26, § 3

a) Let $X$ be a locally compact uniform space, $G$ a group of homeomorphisms of $X$, and $T$ the closure of $G$ in $\mathscr{C}_s(X; X)$. Show that $T$ is compact with respect to the topology of pointwise convergence and is a group of homeomorphisms if and only if $G$ is relatively compact in $\mathscr{C}_c(X; X)$. [Use Exercises 12 and 24 and the Corollary to Theorem 4 (no. 5).]

b) Let $G$ be a locally compact, non-compact topological group and let $X$ be its one-point compactification; $G$ can be identified with a group of homeomorphisms of $X$. Show that $G$ is not equicontinuous, although its closure in $\mathscr{C}_s(X; X)$ is compact.