Definition 2

Tag 0C1B, permanent. Cite it as [Bourbaki, Tag 0C1B].

General Topology, chapter IX, § 1. Generation of a uniformity by a family of pseudometrics; uniformizable spaces

Given a pseudometric $f$ on a set $X$ , the uniformity defined by $f$ is the uniformity on $X$ which has as a fundamental system of entourages the family of sets $\overline{f}^{-1}([0, a])$ , where $a$ runs through the set of all real numbers $> 0$ .

Two pseudometrics on $X$ are said to be equivalent if they define the same uniformity.

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

Cited by

Remark 2, § 1
Definition 2, § 2

Cites

Nothing.