Direct sum of topological groups

In mathematics, a topological group "G" is called the topological direct sum of two subgroups "H"1 and "H"2 if
the map
is a topological isomorphism.
More generally, "G" is called the direct sum of a finite set of subgroups formula_2 of the map
Note that if a topological group "G" is the topological direct sum of the family of subgroups formula_4 then in particular, as an abstract group (without topology) it is also the direct sum (in the usual way) of the family formula_4.
Topological direct summands.
Given a topological group "G", we say that a subgroup "H" is a topological direct summand of "G" (or that splits topologically from "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 formula_7 is the natural inclusion and formula_8 is the natural projection.
Examples.
Suppose that formula_9 is a locally compact abelian group that contains the unit circle formula_10 as a subgroup. Then formula_10 is a topological direct summand of "G". The same assertion is true for the real numbers formula_12