Corollary 3

Let G and $G'$ be topological groups, and let $\varphi : G\rightarrow G'$ be a continuous, open and surjective homomorphism. Suppose that the kernel H of $\varphi$ is discrete. Then, for the operation of H in G by right translations, $\varphi$ makes G into a principal covering of $G'$ of group H.

If moreover the group G is connected, H is contained in the center of G.

The homomorphism $\varphi$ induces a topological group isomorphism of $G/H$ onto $G'$ (TG, III, p. 16, prop. 24), whence the first assertion by Corollary 2. Suppose the group G connected. For every $h\in H$, the continuous mapping of G into H defined by $g\mapsto ghg^{-1}$ is constant, with value $h$. The group H is therefore contained in the center of G.