Exercise 8, § 6

(a) Let $J$ be the Polish space $\mathbf{N}^\mathbf{N}$; now prove that for every Souslin space $X$, there exists a continuous surjective mapping from $J$ onto $X$ (reduce to the case where $X$ is Polish and consider a sieve of $X$ by a sieve $(C_n)$ where all the $C_n$ are infinite).

(b) Let $X$ be a metrizable space; now prove that for every Souslin subspace $A$ of $X$, there exists a closed subset $F$ of $J \times X$ such that $A = \mathrm{pr}_2(F)$ (use $a$).

(c) Let $X$ be a metrizable space of countable type; let $L$ be the space $J \times X$, $F$ a closed subset of $J \times L$ such that, for every closed subset $M$ of $L$, there exists $z \in J$ for which $F(z) = M$ (Exerc. 7 a)). Now prove that for every Souslin subspace $S$ of $X$, there exists $z \in J$ such that $\mathrm{pr}_2(F(z)) = S$. Deduce that, if $X = J$, the set $T$ of the $z \in J$ such that $z \in \mathrm{pr}_2(F(z))$ is Souslin, but that $J - T$ is not Souslin (same reasoning as in Exerc. 7 b)).¹