Remark 1
Tag 0C5J, permanent. Cite it as [Bourbaki, Tag 0C5J].
General Topology, chapter IX, § 4. Normal spaces
Proposition 2 gives another necessary condition for metrizability; but this condition, even in conjunction with all the necessary conditions given in § 2, does not give a set of sufficient conditions for metrizability (cf. Exercise 6 and § 5, Exercise 10).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Nothing.