Remark 1

Tag 0BG9, permanent. Cite it as [Bourbaki, Tag 0BG9].

General Topology, chapter II, § 4. Relations between uniform spaces and compact spaces

A non-Hausdorff quasi-compact space is not necessarily uniformizable, since it need not satisfy axiom $(O_{III})$ (cf. Chapter I, § 9, no. 2); for example most of the non-Hausdorff quasi-compact spaces which appear in algebraic geometry do not satisfy axiom $(O_{III})$ (cf. Exercise 2).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Chapter I, § 9, no. 2
Exercise 2