Exercise 6, § 6

a) Let $J$ be the Polish space $\mathbf{N}^\mathbf{N}$. Show that if $X$ is any Souslin space, there is a continuous surjection of $J$ onto $X$ [reduce to the case where $X$ is Polish and consider a sifting of $X$ by a sieve $(C_n)$ where all the $C_n$ are infinite].

b) Let $X$ be a metrizable space. Show that if $A$ is any Souslin subspace of $X$, there is 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 $L = J \times X$ and let $F$ be a closed subset of $J \times L$ such that for each closed subset

M of L there exists $z \in J$ such that $F(z) = M$ [Exercise 5 a)]. Show that, for each 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 all $z \in J$ such that $z \in \mathrm{pr}_2(F(z))$ is a Souslin set, but that $J - T$ is not a Souslin set [same reasoning as in Exercise 5 b)] (*).