Corollary
Tag 0B8O, permanent. Cite it as [Bourbaki, Tag 0B8O].
General Topology, chapter I, § 9. Compact spaces and locally compact spaces
In a locally compact space every point has a fundamental system of compact neighbourhoods.
For the intersection of a closed neighbourhood of and a compact neighbourhood of is a compact neighbourhood of (no. 3, Proposition 3).
There exist non-Hausdorff topological spaces in which every point has a fundamental system of compact neighbourhoods (Exercise 5).
The Corollary to Proposition 9 may be generalized as follows:
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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