Corollary 2
Tag 0BG1, permanent. Cite it as [Bourbaki, Tag 0BG1].
General Topology, chapter II, § 4. Relations between uniform spaces and compact spaces
Every locally compact space is uniformizable.
For by Alexandroff’s theorem (Chapter I, § 9, no. 8, Theorem 4) a locally compact space is homeomorphic to a subspace of a compact space.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.