Exercise 20, § 5

Let $f$ be a homeomorphism of a subset $A$ of a complete metric space $X$ onto a subset $A'$ of a complete metric space $X'$. Show that there exists an extension $\bar{f}$ of $f$ to a set $B$ which is a countable intersection of open sets in $X$, such that $f$ is a homeomorphism of $B$ onto a set $B'$ which is a countable intersection of open sets in $X'$ (apply Exercise 19 to $f$ and to the inverse homeomorphism $g$).