In mathematics, a topological group G is called the topological direct sum of two subgroups H1 and H2 ifthe map is a topological isomorphism. More generally, G is called the direct sum of a finite set of subgroups of the map Note that if a topological group G is the topological direct sum of the family of subgroups then in particular, as an abstract group (without topology) it is also the direct sum (in the usual way) of the family .