Remark 2

Let G and $G'$ be topological groups and let $\varphi : G\rightarrow G'$ be a continuous open homomorphism. If the group $G'$ is connected, the homomorphism $\varphi$ is surjective. In fact, $\varphi (G)$ is an open, therefore closed, subgroup of $G'$, and consequently equal to $G'$.