Proposition 7

The product and the sum of a countable family of Souslin spaces are Souslin spaces.

For each integer $n$, let $X_n$ be a metrizable space, $P_n$ a Polish space, and $f_n$ a continuous mapping of $P_n$ onto $X_n$. The product (resp. sum) of the spaces $P_n$ is Polish (no. 1, Proposition 1), and the image of this space under the mapping which is the product of the $f_n$ (resp. the mapping which agrees with $f_n$ on $P_n$ for all $n$) is the product (resp. sum) of the spaces $X_n$; the latter is metrizable, and is therefore a Souslin space.