Exercise 19, § 6

Let $X, X'$ be two metrizable spaces and let $f$ be an almost open mapping (§ 5, Exercise 24) of $X$ into $X'$. Show that if $B'$ is any Borel set in $X'$, then $f^{-1}(B')$ is almost open in $X$ (cf. § 5, Exercise 6). Deduce that if $g$ is a Borel mapping of $X'$ into a metrizable space $X''$, then $g \circ f$ is almost open. Give an example of a continuous mapping $f$ and an almost open mapping $g$ such that $g \circ f$ is not almost open [cf. Exercise 10 b)].