Lemma 1

Let $(A_n)$ be a sequence of subsets of a Hausdorff topological space $X$. Then the restriction of the diagonal mapping $f : X \to X^\mathbf{N}$ to the subspace $\bigcap_n A_n$ of $X$ is a homeomorphism of $\bigcap_n A_n$ into a closed subspace of $\prod_n A_n$.

This image is the intersection of $\prod_n A_n$ and the diagonal $\Delta = f(X)$, which is closed in $X^\mathbf{N}$ because $X$ is Hausdorff (Chapter I, $§ 8,$ no. 1); and $f$ is a homeomorphism of $X$ onto $\Delta$.

With the hypotheses of Proposition 3, $\prod_n A_n$ is a Polish space (Proposition 1), hence $\bigcap_n A_n$ is Polish by Lemma 1 and Proposition 1.