Definition 1

Tag 0C2C, permanent. Cite it as [Bourbaki, Tag 0C2C].

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

A metric on a set $X$ is a finite pseudometric $d$ on $X$ such that the relation $d(x, y) = 0$ implies $x = y$ . A metric space is a set $X$ endowed with the structure defined by a given metric on $X$ .

A metric space $X$ is always considered as carrying the uniformity and the topology defined by the given metric on $X$ .

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

Cited by

Example 2, § 2

Cites

Nothing.