Exercise 13, § 3

Let $X, Y, Z$ be three separated topological spaces. a) Now prove that whatever $u_0 \in \mathscr{C}(X; Y)$ and $v_0 \in \mathscr{C}(Y; Z)$ may be, the mappings $v \mapsto v \circ u_0$ of $\mathscr{C}_c(Y; Z)$ into $\mathscr{C}_c(X; Z)$ and $u \mapsto v_0 \circ u$ of $\mathscr{C}_c(X; Y)$ into $\mathscr{C}_c(X; Z)$ are continuous.

b) Suppose that every point of $Y$ (resp. $Z$) admits a countable fundamental system of neighbourhoods, and that there exists in $X$ a countable dense subset. Let $H$ be a compact subset of $\mathscr{C}_c(X; Y)$; prove that the mapping $(u, v) \mapsto v \circ u$ of $H \times \mathscr{C}_c(Y; Z)$ into $\mathscr{C}_c(X; Z)$ is continuous. (One will first prove that for every compact subset $K$ of $X$, $H(K)$ is a compact subset of $Y$, using $X$, p. 48, exerc. 7 and $X$, p. 49, exerc. 9 a)).

c) Prove that the mapping $(u, v) \mapsto v \circ u$ of $\mathscr{C}_c(\mathbf{Q}; \mathbf{Q}) \times \mathscr{C}_c(\mathbf{Q}; \mathbf{Q})$ into $\mathscr{C}_c(\mathbf{Q}; \mathbf{Q})$ is not continuous at any point.