Exercise 20, § 6

Let $X, X'$ be two metrizable spaces. For every countable even (resp. odd) ordinal $\alpha$, one says that a mapping $f$ from $X$ into $X'$ is Borel of class $\alpha$ if, for every closed subset $F'$ of $X'$, $f^{-1}(F')$ belong to $\mathfrak{F}_\alpha(X)$ (resp. $\mathfrak{G}_\alpha(X)$) (IX, p. 119, exerc. 6); then, for every open subset $G'$ of $X'$, $f^{-1}(G')$ belong to $\mathfrak{G}_\alpha(X)$ (resp. $\mathfrak{F}_\alpha(X)$). The Borel functions of class 0 are the continuous functions. $\alpha)$ For the characteristic function $\varphi_A$ of a subset $A$ of $X$ to be of class $\alpha$, it is necessary and sufficient that $A$ belong to $\mathfrak{F}_\alpha \cap \mathfrak{G}_\alpha$. b) Let $f$ be a mapping of class $\alpha$ from $X$ into $X'$; prove that for every subset $B' \in \mathfrak{F}_\beta(X')$ (resp. $B' \in \mathfrak{G}_\beta(X')$), $f^{-1}(B')$ belong to $\mathfrak{F}_{\alpha+\beta}(X)$ (resp. $\mathfrak{G}_{\alpha+\beta}(X)$). (Reason by transfinite induction on $\beta$.) c) Let $f$ be a mapping of class $\alpha$ from $X$ into $X'$, $g$ a mapping of class $\beta$ from $X'$ into a metrizable space $X''$; prove that $g \circ f$ is a mapping of class $\alpha + \beta$. d) For every countable even (resp. odd) ordinal $\alpha$, let $(A_n)$ be a sequence of subsets of $X$ belonging to $\mathfrak{G}_\alpha$ (resp. $\mathfrak{F}_\alpha$) and forming a covering of $X$; if a mapping $f$ from $X$ into $X'$ is such that its restriction to each of the $A_n$ is of class $\alpha$, then $f$ is of class $\alpha$. Analogous result for a finite sequence of subsets of $X$ belonging to $\mathfrak{F}_\alpha$ (resp. $\mathfrak{G}_\alpha$) and forming a covering of $X$.