Direct sum of topological groups

In mathematics, a topological group is called the topological direct sum of two subgroups and if the map
is a topological isomorphism, meaning that it is a homeomorphism and a group isomorphism.

Definition

More generally, is called the direct sum of a finite set of subgroups of the map
is a topological isomorphism.
If a topological group is the topological direct sum of the family of subgroups then in particular, as an abstract group it is also the direct sum of the family

Topological direct summands

Given a topological group we say that a subgroup is a topological direct summand of if and only if there exist another subgroup such that is the direct sum of the subgroups and
A the subgroup is a topological direct summand if and only if the extension of topological groups
splits, where is the natural inclusion and is the natural projection.

Examples

Suppose that is a locally compact abelian group that contains the unit circle as a subgroup. Then is a topological direct summand of The same assertion is true for the real numbers.