Proposition 23

A homomorphism f of a topological group G into a topological group $G'$ is continuous on G if and only if it is continuous at one point of G.

Suppose $f$ is continuous at a point $a \in G$; then if $V'$ is any neighbourhood of $f(a)$, $V = \overline{f^{-1}}(V')$ is a neighbourhood of $a$. Hence if $x$ is any point of $G$, we have

and therefore $f$ is continuous at $x$.

A continuous homomorphism of a topological group $G$ into a topological group $G'$ is also called a morphism of $G$ into $G'$ for the topological group structures.

Let $f$ be a continuous homomorphism of a topological group $G$ into a topological group $G'$; the inverse image $H = \overline{f^{-1}}(e')$ of the identity element $e'$ of $G'$ is a normal subgroup of $G$, and $f(G)$ is a subgroup of $G'$. Consider the canonical factorization $f = \psi \circ \dot{f} \circ \varphi$, where $\varphi$ is the canonical mapping $G \to G/H$, $\psi$ is the canonical injection $f(G) \to G'$ and lastly $\dot{f}$ is a continuous bijective homomorphism of the quotient group $G/H$ onto the subgroup $f(G)$ (Chapter I, § 3, no. 5); $\dot{f}$ is said to be the bijective homomorphism associated with $f$. In general, $\dot{f}$ is not an isomorphism of topological groups.

For example, let $G'$ be a non-discrete topological group, and $G$ the topological group obtained by giving $G'$ the discrete topology; then the identity mapping of $G$ into $G'$ is a continuous bijective homomorphism, but is not bicontinuous.