A local isomorphism between local groups 
and 
is represented by a homeomorphism 
between open
neighborhoods of their identity elements.
Both 
and 
must preserve the identity element. A restricted
product or inverse is defined on either side iff the corresponding
product or inverse is defined on the other, and its value must be preserved by the
maps (Goldbring 2010, p. 1274). Representatives that agree on a smaller open
neighborhood of the identity element represent
the same local isomorphism. The local groups are called
locally isomorphic when such a local isomorphism exists.
Even for topological groups, local isomorphism does not require an isomorphism of the full groups.
For example, the additive real line and the unit
circle under complex multiplication are locally
isomorphic via the map 
on sufficiently small open
neighborhoods of the identity elements, although
the full groups are not isomorphic.
