Definition 2

Tag 0C2M, permanent. Cite it as [Bourbaki, Tag 0C2M].

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

A metric on a set $X$ is said to be compatible with a uniformity $U$ on $X$ if the uniformity defined by the metric coincides with $U$ . A uniformity on a set $X$ is said to be metrizable if there is a metric on $X$ compatible with this uniformity. A uniform space is said to be metrizable if its uniformity is metrizable.

Distinct metrics can be compatible with the same uniformity; they are then equivalent (§ 1, no. 2, Definition 2).

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

Cited by

Nothing.

Cites

§ 1, no. 2, Definition 2