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 $$\begin{align} H_1\times H_2 &\longrightarrow G \\ (h_1,h_2)    &\longmapsto     h_1 h_2 \end{align} $$ is a topological isomorphism, meaning that it is a homeomorphism and a group isomorphism.

Definition
More generally, $$G$$ is called the direct sum of a finite set of subgroups $$H_1, \ldots, H_n$$ of the map $$\begin{align} \prod^n_{i=1} H_i &\longrightarrow G \\ (h_i)_{i\in I}   &\longmapsto     h_1 h_2 \cdots h_n \end{align} $$ is a topological isomorphism.

If a topological group $$G$$ is the topological direct sum of the family of subgroups $$H_1, \ldots, H_n$$ then in particular, as an abstract group (without topology) it is also the direct sum (in the usual way) of the family $$H_i.$$

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 $$K \leq G$$ 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 $$0 \to H\stackrel{i}{{} \to {}} G\stackrel{\pi}{{} \to {}} G/H\to 0$$ splits, where $$i$$ is the natural inclusion and $$\pi$$ is the natural projection.

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