Exercise 23, § 3

Let $X$ be a compact space and let $T$ be a subset of $C(X; X)$ which is stable under the law of composition $(u, v) \to u \circ v$. Suppose that $T$ is endowed with a topology $\mathscr{T}$ finer than that of pointwise convergence and with respect to which $T$ is locally compact; also suppose that, for every countable stable subset $S$ of $T$, the restriction to $S \times X$ of the mapping $\pi : (u, x) \to u(x)$ of $T \times X$ into $X$ is continuous. Show that, under these conditions, $\pi$ is continuous on $T \times X$. [Let $V$ be a compact subset of $T$ (with respect to $\mathscr{T}$); show that, for each countable subset $D$ of $V$, the closure of $D$ with respect to $\mathscr{T}$ is contained in the closure of $D$ with respect to the topology of uniform convergence, by using Corollary 1 of Theorem 3 (no. 4). Deduce that $V$ is relatively compact in $\mathscr{C}_u(X; X)$ (Chapter II, § 4, Exercise 6), and hence equicontinuous.]