Proposition 2

Let G be a connected topological group, let $G'$ be a group and let $f: V\rightarrow G'$ be a local homomorphism of G into $G'$, where V is a connected neighbourhood of the identity element of G. Then there exist a connected topological group H, a morphism of topological groups $p: H\rightarrow G$ such that $(H, p)$ is a covering of G, and a group homomorphism $\varphi : H\rightarrow G'$ such that the set of $y\in \overset{-1}{p}(V)$ satisfying $f(p(y)) =\varphi (y)$ is a neighbourhood of the identity element in H.

If $G'$ is a topological group and if the mapping $f$ is continuous, such a homomorphism $\varphi$ is continuous.