Definition 1

Given a topological group G and a group $G'$, a local homomorphism of G into $G'$ means a mapping $f$ of a neighbourhood V of the identity element of G into $G'$ such that, for every pair of points $x,y$ of V such that $xy\in V$, one has $f(xy) =f(x)f(y)$.

If $G'$ is a topological group and if $f$ is a continuous mapping, one says that $f$ is a continuous local homomorphism (or local morphism of topological groups).

If G and $G'$ are topological groups, the notion of local isomorphism of G onto $G'$ has been defined (TG, III, p. 6, Def. 2). A local isomorphism of G onto $G'$ is a homeomorphism $f$ of a neighbourhood V of the identity element of G onto a neighbourhood $V'$ of the identity element of $G'$ such that $f$ and the inverse mapping of $f$ are local homomorphisms.