Exercise 7, § 1

Let $X, Y$ be two Hausdorff uniform spaces. For each continuous mapping $f : X \to Y$, let $G(f) \subset X \times Y$ be the graph of $f$; $G(f)$ is closed in $X \times Y$ (Chapter I, § 8, no. 1, Proposition 2, Corollary 2); hence $f \to G(f)$ is an injective mapping of $\mathscr{C}(X; Y)$ into the set $\tilde{\mathfrak{F}}(X \times Y)$ of non-empty closed subsets of $X \times Y$.

a) Show that the mapping $f \to G(f)$ of $\mathscr{C}_u(X; Y)$ into $\tilde{\mathfrak{F}}(X \times Y)$ is uniformly continuous when $\tilde{\mathfrak{F}}(X \times Y)$ carries the uniformity defined in Chapter II, § 1, Exercise 5.

b) Let $\Gamma$ be the image of $\mathscr{C}(X; Y)$ in $\tilde{\mathfrak{F}}(X \times Y)$ under the mapping $f \to G(f)$, and let $\varphi : \Gamma \to \mathscr{C}_u(X; Y)$ be the inverse mapping. Show that if $X$ is compact, then $\varphi$ is continuous on $\Gamma$ (argue by contradiction).

c) Take both $X$ and $Y$ to be the compact interval $[0, 1]$ of $\mathbf{R}$. Show that $\varphi$ is not uniformly continuous on $\Gamma$.