Proposition 11

In a Souslin space $X$, every Borel set is a Souslin set.

Let $\mathfrak{I}$ be the set of all subsets $A$ of $X$ such that both $A$ and $\overline{A}$ are Souslin sets. By Proposition 8 of no. 2, $\mathfrak{I}$ is a $\sigma$-algebra. Every closed subset $F$ of $X$ belongs to $\mathfrak{I}$, for both $F$ and $\overline{F}$ are Souslin sets (no. 2, Proposition 5); hence $\mathfrak{I}$ contains all Borel sets of $X$ (cf. no. 6, Theorem 2, Corollary).