Proposition 6
Tag 0C2R, permanent. Cite it as [Bourbaki, Tag 0C2R].
General Topology, chapter IX, § 2. Metric spaces and metrizable spaces
Every point of a metrizable space has a countable fundamental system of neighbourhoods.
More generally:
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Cites
Nothing.