Direct sum of topological groups


In mathematics, a topological group G is called the topological direct sum of two subgroups H1 and H2 if
the 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 it is also the direct sum of the family.

Topological direct summands

Given a topological group G, we say that a subgroup H is a topological direct summand of G if and only if there exist another subgroup K ≤ G such that G is the direct sum of the subgroups H and K.
A the subgroup H 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