Exercise 17, § 6

a) Let $X, X'$ be two metrizable spaces, $X'$ being of countable type, and let $(U'_n)$ be a countable base for the topology of $X'$. Let $\alpha$ be an even (resp. odd) countable ordinal. If a mapping $f : X \to X'$ is such that $f^{-1}(U'_n)$ belongs to $\mathscr{G}_\alpha(X)$ [resp. $\mathfrak{F}_\alpha(X)$] for each $n$, then $f$ is a Borel mapping of class $\alpha$.

b) Let $(X'_n)$ be a sequence of metrizable spaces of countable type. A mapping $f = (f_n) : X \to \prod_n X'_n$ is of class $\alpha$ if and only if each $f_n$ is of class $\alpha$ [use a)]. Deduce that, if $X$ is of countable type, the finite real-valued functions of class $\alpha$ on $X$ form a ring.

c) Let $X, X'$ be two metrizable spaces of countable type, and let $\alpha$ be an even (resp. odd) countable ordinal. If a mapping $f : X \to X'$ is of class $\alpha$, show that its graph belongs to $\mathfrak{F}_\alpha(X \times X')$ [resp. $\mathscr{G}_\alpha(X \times X')$] [use b) and Exercise 16 b)].