习题 11, § 4

令X为豪斯多夫均匀空间，$\mathfrak{U}$其均匀性，$\mathfrak{F}(X)$集合为X的所有非空闭子集。 令 $\mathfrak{F}(X)$ 带有由 § 1、习题 5 a) 中定义的均匀性 $\tilde{\mathfrak{U}}$ 引起的均匀性。 a) 证明如果 X' 是预紧的，那么 $\mathfrak{F}(X)$ 也是预紧的。 [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].