Exercise 16, § 1

Let $X$ be a complete Hausdorff uniform space, and let $\mathscr{U}$ be the uniformity of $X$.

a) Let $A$ be a subset of $X$ which is the union of a sequence $(F_n)$ of closed sets, and let $x \notin A$. Show that there is a continuous finite real-valued function $f \geq 0$ on $X$, such that $f(x) = 0$ and $f(y) > 0$ for all $y \in A$. (If $g_n$ is a continuous mapping of $X$ into $[0, 1]$ such that $g_n(x) = 0$ and $g_n(y) = 1$ for all $y \in F_n$, consider the function

b) Let $h \geq 0$ be a continuous real-valued function on $X$, and let $V$ be the open set $\overline{h}([0, +\infty[)$. Let $\mathscr{U}'$ be the uniformity on $V$ which is the least upper bound of the uniformity induced by $\mathscr{U}$ and the uniformity defined by the pseudometric

Show that $V$ is complete with respect to $\mathscr{U}'$.

c) Let $B$ be a subset of $X$ which is the intersection of a family of sets $(A_\lambda)$, each of which is the union of a countable family of closed sets. Show that there exists a uniformity on $B$ which is compatible with the topology induced on $B$ by the topology of $X$, and such that $B$ is complete with respect to this uniformity [use b)].