Theorem 1
Tag 0C2N, permanent. Cite it as [Bourbaki, Tag 0C2N].
General Topology, chapter IX, § 2. Metric spaces and metrizable spaces
A uniformity is metrizable if and only if it is Hausdorff and the filter of entourages of the uniformity has a countable base.
The condition is necessary, for (with the notation of no. 2) the entourages form a base of the filter of entourages of the uniformity of a metric space.
The condition is sufficient, for, if it is satisfied, the uniformity under consideration is defined by a single pseudometric, by Proposition 2 of § 1, no. 4; since the uniformity is Hausdorff, this pseudometric is a metric.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
Cited by
- Theorem 1 (Mittag-Leffler), § 3
- Definition 3, § 2
- Proposition 16, § 2
- Proposition 1, § 3
- Exercise 12, § 4
- Proposition 1, § 6
- Proposition 9, § 6
- Corollary, § 3