title: Direct sum of topological groups
text: # 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
  
  ${\begin{aligned}H_{1}\times H_{2}&\longrightarrow G\\(h_{1},h_{2})&\longmapsto h_{1}h_{2}\end{aligned}}$
  
  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
  
  ${\begin{aligned}\prod _{i=1}^{n}H_{i}&\longrightarrow G\\(h_{i})_{i\in I}&\longmapsto h_{1}h_{2}\cdots h_{n}\end{aligned}}$
  
  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 $\mathbb {R}$
  
  