Exercise 18, § 6

Let $X$ be a Souslin space, $X'$ a metrizable space of countable type and let $f$ be a mapping of $X$ into $X'$.

a) Show that if the graph $G$ of $f$ is a Souslin set in $X \times X'$, then $f$ is a Borel function and therefore [Exercise 17 c)] $G$ is a Borel set (use Theorem 2, Corollary).

b) Show that if $f$ is an injective Borel mapping and if $X'$ is a Souslin space, then the image under $f$ of any Souslin set in $X$ is a Souslin set in $X'$ (use Exercise 17 c)). If $f$ is bijective, the inverse mapping is also a Borel mapping.