Proposition 24

Let $f$ be a continuous homomorphism of a topological group $G$ into a topological group $G'$. Then the following three statements are equivalent: a) $f$ is a strict morphism. b) The image under $f$ of every open set in $G$ is an open set in $f(G)$. c) The image under $f$ of every neighbourhood of the identity element in $G$ is a neighbourhood of the identity element in $G'$.

In view of Lemma 2 of no. 4, the equivalence of a) and b) follows immediately from the definitions (Chapter I, § 5, no. 3, Proposition 5). The equivalence of b) and c) is a particular case of Proposition 15 of no. 5, if we observe that $G$ operates continuously on $f(G)$ by the external law $(s, f(t)) \to f(st)$.