Exercise 24, § 3

Let $X$ be a locally compact space, $T$ a group of homeomorphisms of $X$, $\mathscr{T}$ a locally compact topology on $T$, which is finer than the topology of pointwise convergence and for which condition (A) of Exercise 21 is satisfied. Show that the mapping $(u, x) \to u(x)$ of $T \times X$ into $X$ is continuous with respect to $\mathscr{T}$. [Extend the elements of $T$ to homeomorphisms of the one-point compactification $X'$ of $X$; then, by using Exercise 22 d), show that the result of Exercise 23 is applicable.]