Proposition 3

Tag 0BK0, permanent. Cite it as [Bourbaki, Tag 0BK0].

General Topology, chapter III, § 3. Uniform structures on groups

Every continuous homomorphism $f$ of a topological group $G$ into a topological group $G'$ is uniformly continuous when considered as a mapping of $G_d$ into $G'_d$ (or of $G_s$ into $G'_s$ ).

For if $V'$ is a neighbourhood of the identity element in $G'$ , and $V = \overline{f}^{-1}(V')$ , then the relation $yx^{-1} \in V$ implies

f(y)(f(x))^{-1} = f(yx^{-1}) \in V'.

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

Cited by

Proposition 5, § 3
Theorem 2, § 3
Exercise 3, § 7

Cites

Nothing.