More generally, ${\displaystyle G}$  is called the direct sum of a finite set of subgroups ${\displaystyle H_{1},\ldots ,H_{n}}$  of the map

If a topological group ${\displaystyle G}$  is the topological direct sum of the family of subgroups ${\displaystyle H_{1},\ldots ,H_{n}}$  then in particular, as an abstract group (without topology) it is also the direct sum (in the usual way) of the family ${\displaystyle H_{i}.}$ 

Given a topological group ${\displaystyle G,}$  we say that a subgroup ${\displaystyle H}$  is a topological direct summand of ${\displaystyle G}$  (or that splits topologically from ${\displaystyle G}$ ) if and only if there exist another subgroup ${\displaystyle K\leq G}$  such that ${\displaystyle G}$  is the direct sum of the subgroups ${\displaystyle H}$  and ${\displaystyle K.}$ 

A the subgroup ${\displaystyle H}$  is a topological direct summand if and only if the extension of topological groups

Suppose that ${\displaystyle G}$  is a locally compact abelian group that contains the unit circle ${\displaystyle \mathbb {T} }$  as a subgroup. Then ${\displaystyle \mathbb {T} }$  is a topological direct summand of ${\displaystyle G.}$  The same assertion is true for the real numbers ${\displaystyle \mathbb {R} }$ ^[2]

• Complemented subspace
• Direct sum – Operation in abstract algebra composing objects into "more complicated" objects
• Direct sum of modules – Operation in abstract algebra