Corollary 3
Tag 0247, permanent. Cite it as [Bourbaki, Tag 0247].
Topologie algébrique, chapter IV, § 6. Espaces classifiants
Let B be a paracompact topological space. If B is homotopic to a point, every locally trivial fibre space with base B is trivializable.
Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.
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