Exercise 6, § 2

Let $X$ be a uniform space, $\mathfrak{U}$ its uniformity, $\tilde{\mathfrak{U}}$ the corresponding uniformity on $\mathfrak{P}(X)$ ($§ 1$, Exercise 5). a) Show that the mapping $x \to \{x\}$ is an isomorphism of the uniform space $X$ onto a subspace of the uniform space $\mathfrak{P}(X)$. b) Show that the uniformity induced by $\tilde{\mathfrak{U}}$ on the set $\mathfrak{F}(X)$ of non-empty closed subsets of $X$ is Hausdorff, and that $\tilde{\mathfrak{U}}$ is the inverse image under the mapping $M \to \overline{M}$ of the uniformity induced by $\tilde{\mathfrak{U}}$ on $\mathfrak{F}(X)$. c) Show that the mapping $(M, N) \to M \cup N$ of $\mathfrak{P}(X) \times \mathfrak{P}(X)$ into $\mathfrak{P}(X)$ is uniformly continuous. d) Let $Y$ be another uniform space. If $f : X \to Y$ is a uniformly continuous mapping, show that the mapping $M \to f(M)$ of $\mathfrak{P}(X)$ into $\mathfrak{P}(Y)$ is uniformly continuous. e) Let $\mathfrak{B}$ be a compact subset of the set $\mathfrak{F}(X)$ of non-empty closed subsets of $X$, $\mathfrak{F}(X)$ carrying the topology induced by the topology $\mathfrak{T}(\tilde{\mathfrak{U}})$ induced by $\tilde{\mathfrak{U}}$. Show that the set $\bigcup_{M \in \mathfrak{B}} M$ is closed in $X$.