Exercise 5, § 1

a) Let $X$ be a uniform space, $\mathfrak{U}$ its uniformity; for each entourage $V$ of $X$, let $\tilde{V}$ be the subset of $\mathfrak{P}(X) \times \mathfrak{P}(X)$ consisting of all pairs $(M, N)$ of subsets of $X$ such that both $M \subset V(N)$ and $N \subset V(M)$. Show that the sets $\tilde{V}$ form a fundamental system of entourages of a uniformity $\tilde{\mathfrak{U}}$ on $\mathfrak{P}(X)$.

b) On the set $\mathfrak{P}_0(X)$ of non-empty subsets of $X$, show that the topology induced by the topology $\mathfrak{T}(\tilde{\mathfrak{U}})$ induced by $\tilde{\mathfrak{U}}$ is finer than the topology $\mathfrak{T}_\Phi$ (Chapter I, § 2, Exercise 7).

c) If $X$ has at least two points and if $\mathfrak{U}$ is Hausdorff, show that the topology induced by $\mathfrak{T}(\tilde{\mathfrak{U}})$ on $\mathfrak{P}_0(X)$ is never coarser than $\mathfrak{T}_\Omega$. The topology induced by $\mathfrak{T}(\tilde{\mathfrak{U}})$ on the set $\mathfrak{F}(X)$ of non-empty closed subsets of $X$ is finer than the topology induced by $\mathfrak{T}_\Omega$ on this set if and only if, for each closed subset $A$ of $X$, the sets $V(A)$ form a fundamental system of neighbourhoods of $A$ as $V$ runs through the set of entourages of $X$.

d) Show that on the quotient set $X/R$ defined in Chapter I, § 11, Exercise 18, the quotient topology, and the topologies defined by $\mathfrak{T}_\Omega, \mathfrak{T}_\Phi$ and $\mathfrak{T}(\tilde{\mathfrak{U}})$ (where $\mathfrak{U}$ is the usual uniformity on $\mathbf{R}^2$) are all four distinct.