Exercise 25, § 1

Let $X$ be a separated and complete uniform space, $\mathscr{U}$ its uniform structure. a) Let $h$ be a numerical function $\geq 0$, continuous in $X$, and let $V$ be the open set $h^{-1}(0, +\infty)$. Consider on $V$ the uniform structure $\mathscr{U}'$, supremum of the structure induced by $\mathscr{U}$ and of the uniform structure defined by the distance

Now prove that $V$ is complete for $\mathscr{U}'$. b) Let $B$ be a subset of $X$, intersection of a family $(A_\lambda)$ of sets each of which is a union of a countable family of closed sets in $X$. Now prove that there exists on $B$ a uniform structure compatible with the topology induced by that of $X$, and for which $B$ is complete (use a) and Exer. 19 a) of IX, p. 88).