Proposition 7

Tag 0BE2, permanent. Cite it as [Bourbaki, Tag 0BE2].

General Topology, chapter II, § 2. Uniformly continuous functions

Let $f = (f_i)$ be a mapping of a uniform space $Y$ into a product uniform space $X = \prod_{i \in I} X_i$ . Then $f$ is uniformly continuous if and only if each $f_i$ is uniformly continuous.

Since $f_i = pr_i \circ f$ , this is a particular case of Proposition 4 of no. 3.

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

Cited by

Corollary, § 2
§ 1. Real number space R^n

Cites

Proposition 4