Exercise 24, § 2

a) Let $X$ be a complete metric space, and let $A$ be a subset of $X$ which is the intersection of a countable family of open sets. Show that there exists a metric on $A$ with respect to which $A$ is a complete metric space, and which defines on $A$ the topology induced by that of $X$ and also defines a uniformity finer than that induced by the uniformity of $X$ [cf. $§ 1$, Exercise 16 b) and c)].

b) Conversely, let $X$ be a metrizable space and let $A$ be a subset of $X$ such that there exists a metric $d$ on $A$ which is compatible with the topology induced by that of $X$ and for which $A$ is a complete metric space. Show that $A$ is a countable intersection of open sets in $X$. [For each integer $n > 0$, consider the set $G_n$ of points $x \in \overline{A}$ which have an open neighbourhood $U$ such that the diameter of $A \cap U$ (with respect to $d$) is $\leqslant 1/n$.]