Definition 1

Tag 0C88, permanent. Cite it as [Bourbaki, Tag 0C88].

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

A topological space $X$ is said to be Polish if it is metrizable of countable type ( $§ 2$ , no. 8) and if there is a metric, compatible with the topology of $X$ , with respect to which $X$ is complete.

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

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