Exercise 21, § 6

a) Let $X, X'$ be two metrizable spaces, $X'$ being of countable type, and let $(U'_n)$ be a countable basis for the topology of $X'$. Let $\alpha$ be an even (resp. odd) countable ordinal; if a mapping $f$ of $X$ into $X'$ is such that $f^{-1}(U'_n)$ belongs to $\mathfrak{G}_\alpha(X)$ (resp. $\mathfrak{F}_\alpha(X)$), $f$ is of class $\alpha$. b) Let $(X_n)$ be a sequence of metrizable spaces of countable type. In order that a mapping $f = (f_n)$ of $X$ into $\prod_n X'_n$ be of class $\alpha$, it is necessary and sufficient that each of the $f_n$ be of class $\alpha$ (use a)). Deduce that if $X$ is of countable type, the finite numerical functions of class $\alpha$ on $X$ form a ring. c) Let $X, X'$ be two metrizable spaces of countable type, $\alpha$ an even (resp. odd) countable ordinal. If a mapping $f$ of $X$ into $X'$ is of class $\alpha$, now prove that its graph belongs to $\mathfrak{F}_\alpha(X \times X')$ (resp. $\mathfrak{G}_\alpha(X \times X')$) (use b) and Exercise 20 b)).