Proposition 11
Tag 0B8Q, permanent. Cite it as [Bourbaki, Tag 0B8Q].
General Topology, chapter I, § 9. Compact spaces and locally compact spaces
Let X be a locally compact space and F a subset of X such that F \cap K is compact whenever K is a compact subset of X. Then F is closed in X.
In view of Proposition 4 of no. 3, this follows from Proposition 3 a) of § 3, no. 1.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Nothing.