Corollary 1

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

General Topology, chapter I, § 10. Proper mappings

A topological space $X$ is quasi-compact if and only if the mapping $X \to P$ is proper.

Apply a) $\iff$ b) to $X \to P$ .

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

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Nothing.