Proposition 9

Let $X, Y, Z$ be three topological spaces, $Y$ being locally compact. Then the mapping $(u, v) \to v \circ u$ of $\mathscr{C}_c(X; Y) \times \mathscr{C}_c(Y; Z)$ into $\mathscr{C}_c(X; Z)$ is continuous.

We have to show that, for every compact subset $K$ of $X$ and every open subset $U$ of $Z$, the set $R$ of pairs $(u, v)$ such that $v(u(K)) \subset U$ is open in $\mathscr{C}_c(X; Y) \times \mathscr{C}_c(Y; Z)$. Let $(u_0, v_0) \in R$; then $u_0(K)$ is a compact subset of the locally compact space $Y$, contained in the open set $\overline{v_0^{-1}}(U)$, and hence there is a compact neighbourhood $L$ of $u_0(K)$ contained in $\overline{v_0^{-1}}(U)$ (Chapter I, § 9, no. 7, Proposition 10). The set $V$ of all $u \in \mathscr{C}_c(X; Y)$ such that $u(K) \subset L$ is a neighbourhood of $u_0$, and the set $W$ of all $v \in \mathscr{C}_c(Y; Z)$ such that $v(L) \subset U$ is a neighbourhood of $v_0$; furthermore, the relation $(u, v) \in V \times W$ implies $v(u(K)) \subset U$. Hence the result.