Exercise 1, § 5

Let $X$ be a locally compact space, $\mathfrak{F}$ (or $\mathfrak{F}(X)$) the set of closed subsets of $X$. For every compact subset $K$ of $X$, every compact neighborhood $L$ of $K$, and every entourage $V$ of the unique uniform structure of $L$, let $Q(K, L, V)$ be the set of pairs $(A, B)$ of elements of $\mathfrak{F}$ satisfying the two conditions

a) Show that the sets $Q(K, L, V)$ form a fundamental system of entourages for a Hausdorff uniform structure on $\mathfrak{F}$.

b) Let $\mathscr{U}$ be a uniform structure compatible with the topology of $X$. For every compact subset $K$ of $X$ and every entourage $W$ of $\mathscr{U}$, let $P(K, W)$ be the set of pairs $(A, B)$ of $\mathfrak{F}$ satisfying the two conditions

Show that the sets $P(K, W)$ form a fundamental system of entourages for the uniform structure defined in a).

c) Let $X'$ be the Alexandroff compactification of $X$, $\omega$ the point at infinity of $X'$; for every closed subset $A$ of $X$, let $A' = A \cup \{\omega\}$. Show that the mapping $A \mapsto A'$ is an isomorphism of the uniform space $\mathfrak{F}(X)$ defined in a) onto the subspace of $\mathfrak{F}(X')$ formed by the closed subsets containing $\omega$ (make use of b)). Deduce from this that $\mathfrak{F}(X)$ is compact (cf. GT, II, §4, Exer. 15).