Proposition 1

Let G and $G'$ be topological groups and let $p: G\rightarrow G'$ be a group homomorphism. In order that $p$ should make G a covering of $G'$, it is necessary and sufficient that the restriction of $p$ to a suitable neighbourhood of the identity element of G should be a local isomorphism of G onto $G'$.

The condition is necessary by Proposition 3 of TG, III, p. 6. Conversely, suppose that $p$ induces a homeomorphism of an open neighbourhood V of the identity element $e$ of G onto a neighbourhood $V'$ of the identity element of $G'$. The mapping $p$ is then continuous (TG, III, p. 15, Prop. 23) and open (TG, III, p. 16, Prop. 24). The image H of $p$ is a subgroup of $G'$ containing V, hence open and closed in $G'$ (TG, III, p. 7, Corollary). Let N be the kernel of $p$; one has $N\cap V =\{e\}$, therefore N is discrete (loc. cit., Prop. 5). Consequently, $p$ makes G a principal covering of H, with group N (I, p. 100, Cor. 3 of Th. 1). Since H is open and closed in $G'$, the $G'$-space $(G, p)$ is a covering.