Exercise 11, § 4

Let X be a Hausdorff uniform space, $\mathfrak{U}$ its uniformity, $\mathfrak{F}(X)$ the set of all non-empty closed subsets of X. Let $\mathfrak{F}(X)$ carry the uniformity induced by the uniformity $\tilde{\mathfrak{U}}$ defined in § 1, Exercise 5 a). a) Show that if X' is precompact then so is $\mathfrak{F}(X)$. [If $(A_i)$ is a finite covering of X by V-small sets (V symmetric), show that the sets $\tilde{V}(B_k)$ cover $\mathfrak{F}(X)$, where the B_k are all the unions of sets A_i]. b) A point A ∈ $\mathfrak{F}(X)$ is such that every neighbourhood of A in the topology induced by $\mathfrak{T}(\tilde{\mathfrak{U}})$ on $\mathfrak{F}(X)$ contains a neighbourhood of A in the topology induced by $\mathfrak{T}_\Theta$ (Chapter I, § 8, Exercise 12) on $\mathfrak{F}(X)$, if and only if A is precompact. c) Show that the topologies $\mathfrak{T}(\tilde{\mathfrak{U}})$ and $\mathfrak{T}_\Theta$ induce the same topology on the set $\mathfrak{K}(X)$ of compact subsets of X [use b) and Exercise 5 of § 1].