Direct sum of topological groups

In mathematics, a topological group

is called the topological direct sum[1] of two subgroups

if the map

is a topological isomorphism, meaning that it is a homeomorphism and a group isomorphism.

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 (without topology) it is also the direct sum (in the usual way) of the family

Given a topological group

is a topological direct summand of

(or that splits topologically from

) if and only if there exist another subgroup

is the direct sum of the subgroups

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.

Suppose that

is a locally compact abelian group that contains the unit circle

is a topological direct summand of

The same assertion is true for the real numbers