Corollary

Tag 0BG8, permanent. Cite it as [Bourbaki, Tag 0BG8].

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

A uniform space $X$ is compact if and only if it is Hausdorff and complete and can be covered by a finite number of $V$ -small sets, where $V$ is any entourage of $X$ .

This follows from Theorem 3 and Theorem 1 of no. 1.

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

Cited by

Theorem 2 (Borel-Lebesgue), § 2

Cites

Theorem 1
Theorem 3