Proposition 14

Tag 0C32, permanent. Cite it as [Bourbaki, Tag 0C32].

General Topology, chapter IX, § 2. Metric spaces and metrizable spaces

A metric space X is precompact if and only if, for each $\varepsilon > 0$ , there is a finite covering of X by sets of diameter $\leq \varepsilon$ .

If we adjoin the hypothesis that X is complete we have a criterion for compactness of metric spaces.

From Proposition 14 we get a topological criterion for compactness, applicable to metrizable spaces:

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

Cited by

Proposition 15, § 2
Proposition 16, § 2

Cites

Nothing.