Corollary

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

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

The topology on $X$ induced by the coarsest uniformity $\mathfrak{U}$ for which the $f_i$ are uniformly continuous is also the coarsest topology for which the $f_i$ are continuous.

This is an immediate consequence of the definition of the neighbourhoods of a point in this latter topology (Chapter I, § 2, no. 3, Proposition 4).

The general properties of initial structures (Set Theory, Chapter IV, § 2, no. 3, criterion CST 10) imply in particular the following transitivity property:

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

Cited by

§ 2. Uniformly continuous functions
Definition 4, § 2

Cites

*Set Theory*, Chapter IV, § 2, no. 3
Chapter I, § 2, no. 3, Proposition 4