Definition 2

If $G$ and $G'$ are two topological groups, a local isomorphism of $G$ with $G'$ is a homeomorphism $f$ of a neighbourhood $V$ of the identity element of $G$ onto a neighbourhood $V'$ of the identity element of $G'$ which satisfies the following conditions:

1) For each pair $x, y$ of points of $V$ such that $xy \in V$,

2) If g is the mapping inverse to f, then for each pair of points x', y' of V' such that x'y' ∈ V', we have g(x'y') = g(x') g(y'). The mapping g is then a local isomorphism of G' with G. Two topological groups G, G' are said to be locally isomorphic if there exists a local isomorphism of G with G'.

Isomorphic topological groups are evidently locally isomorphic. The converse is false.

\* For example, we shall see in Chapter V, § 1, that the topological groups R and T are locally isomorphic but not isomorphic. \*

If f is a local isomorphism of G with G', then every restriction of f to a neighbourhood of the identity element of G is again a local isomorphism of G with G'.

A local isomorphism of G with G is called a local automorphism of G.

In general, if f is a homeomorphism of a neighbourhood V of the identity element of G onto a neighbourhood V' of the identity element of G' which satisfies condition 1) of Definition 2, f does not necessarily satisfy condition 2) (see Exercise 7). Nevertheless, G and G' are in fact locally isomorphic.