Exercise 12, § 3

Let X be a topological space, Y and Z two uniform spaces, $\mathscr{S}$ a set of subsets of X and $\mathscr{T}$ a set of subsets of Y.

a) Let $u_0 \in \mathscr{C}(X; Y)$. Show that if, for each $A \in \mathscr{S}$, $u_0(A)$ is contained in some set of $\mathscr{T}$, then the mapping $v \to v \circ u_0$ of $\mathscr{C}_{\mathscr{T}}(Y; Z)$ into $\mathscr{C}_{\mathscr{S}}(X; Z)$ is continuous.

b) Let H be a subspace of $\mathscr{C}_{\mathscr{S}}(X; Y)$ and let $v_0 \in \mathscr{C}(Y; Z)$. Show that if $v_0$ is uniformly continuous on $H(A)$ for all $A \in \mathscr{S}$, then the mapping $u \to v_0 \circ u$ of H into $\mathscr{C}_{\mathscr{S}}(X; Z)$ is continuous.

c) Let $u_0 \in \mathscr{C}(X; Y)$ and let $v_0 \in \mathscr{C}(Y; Z)$. Suppose that, for each $A \in \mathscr{S}$, there exists $B \in \mathscr{T}$ and an entourage V of Y satisfying the conditions: (i) $V(u_0(A)) \subset B$, (ii) $v_0$ is uniformly continuous on B. Then the mapping $(u, v) \to v \circ u$ of $\mathscr{C}_{\mathscr{S}}(X; Y) \times \mathscr{C}_{\mathscr{T}}(Y; Z)$ into $\mathscr{C}_{\mathscr{S}}(X; Z)$ is continuous at $(u_0, v_0)$. In particular, the mapping $(u, v) \to v \circ u$ of $\mathscr{C}_u(X; Y) \times \mathscr{C}_u(Y; Z)$ into $\mathscr{C}_u(X; Z)$ is continuous at every point $(u_0, v_0)$ such that $v_0$ is uniformly continuous on Y.

d) Let $v_0$ be the homeomorphism $x \to x^3$ of $\mathbf{R}$ onto itself. Show that the mapping $u \to v_0 \circ u$ of $\mathscr{C}_u(\mathbf{R}; \mathbf{R})$ into itself is not continuous at every point.