Corollary 2

Tag 0C8H, permanent. Cite it as [Bourbaki, Tag 0C8H].

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

Let $X$ and $Y$ be two Polish spaces and let $f : X \to Y$ be a continuous mapping. If $Z$ is a Polish subspace of $Y$ , then $\overline{f}^{-1}(Z)$ is a Polish subspace of $X$ .

For $Z = \bigcap_n Z_n$ , where the $Z_n$ are open subsets of $Y$ ; hence

\overline{f}^{-1}(Z) = \bigcap_n \overline{f}^{-1}(Z_n),

and the sets

\overline{f}^{-1}(Z_n)

are open in

X

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

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