Exercise 6, § 2

Let $X$ be a metric space. If $A$ and $B$ are two non-empty subsets of $X$, put

also put $\sigma(\emptyset, \emptyset) = 0, \sigma(\emptyset, A) = \sigma(A, \emptyset) = +\infty$ for any non-empty subset $A$ of $X$. Show that $\sigma$ is a pseudometric on the set $\mathfrak{P}(X)$ of all subsets of $X$, and that the uniformity defined by $\sigma$ coincides with the uniformity constructed from that of $X$ by the procedure of Chapter II, § 1, Exercise 5.

Suppose that $X$ is bounded. Then $\sigma$ is a metric on the set $\mathfrak{F}(X)$ of non-empty closed subsets of $X$. If, moreover, $X$ is complete, show that $\mathfrak{F}(X)$ is a complete metric space [let $\Phi$ be a Cauchy filter on $\mathfrak{F}(X)$; for each set $\mathscr{X} \in \Phi$, let $S(\mathscr{X})$ be the union of the subsets $A$ of $X$ which belong to $\mathscr{X}$; show that the sets $S(\mathscr{X})$ form a filter base on $X$ as $\mathscr{X}$ runs through $\Phi$, that this filter base has a non-empty set $C$ of cluster points, and that $\Phi$ converges to $C$].