Proposition 3

Tag 0C8C, permanent. Cite it as [Bourbaki, Tag 0C8C].

General Topology, chapter IX, § 6. Polish spaces, Souslin spaces, Borel sets

Let $X$ be a Hausdorff topological space. Then the intersection of a sequence $(A_n)$ of Polish subspaces of $X$ is a Polish subspace.

Let $f$ be the diagonal mapping of $X$ into $X^\mathbf{N}$ ([Set Theory, Chapter II, $§ 5,$ no. 3; recall that $f(x) = (x, x, \ldots)$ where $x_n = x$ for all $n \in \mathbf{N}$ ]. We shall use the following lemma:

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Lemma 1, § 6
Proposition 2, § 6

Cites

Nothing.