Exercise 13, § 3

Let X, Y, Z be three Hausdorff topological spaces.

a) Show that for all $u_0 \in \mathscr{C}(X; Y)$ and all $v_0 \in \mathscr{C}(Y; Z)$, the mappings $v \to v \circ u_0$ of $\mathscr{C}_c(Y; Z)$ into $\mathscr{C}_c(X; Z)$ and $u \to v_0 \circ u$ of $\mathscr{C}_c(X; Y)$ into $\mathscr{C}_c(X; Z)$ are continuous.

b) Suppose that every point of Y (resp. Z) has a countable fundamental system of neighbourhoods and that X has a countable dense subset. Let H be a compact subset of $\mathscr{C}_c(X; Y)$. Show that the mapping $(u, v) \to v \circ u$ of $H \times \mathscr{C}_c(Y; Z)$ into $\mathscr{C}_c(X; Z)$ is continuous. [Start by showing that if K is any compact subset of X, then H(K) is a compact subset of Y, by using Exercises 7 and 9 a).]

c) Show that the mapping $(v, v) \to v \circ u$ 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.