Direct sum of topological groups

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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
topological group In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
$G$ is called the topological direct sum of two
subgroup In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ...
s $H_1$ and $H_2$ if the map $\begin H_1\times H_2 &\longrightarrow G \\ (h_1,h_2) &\longmapsto h_1 h_2 \end$ is a topological isomorphism, meaning that it is a
homeomorphism In the mathematics, mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and Continuous function#Continuous functions between topological spaces, continuous function between topologic ...
and a
group isomorphism In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two ...
.

# Definition

More generally, $G$ is called the direct sum of a finite set of
subgroup In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ...
s $H_1, \ldots, H_n$ of the map $\begin \prod^n_ H_i &\longrightarrow G \\ (h_i)_ &\longmapsto h_1 h_2 \cdots h_n \end$ 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 The direct sum is an Operation (mathematics), operation between Mathematical structure, structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct ...
(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 $0 \to H\stackrel G\stackrel G/H\to 0$ 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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
$\mathbb$ as a subgroup. Then $\mathbb$ is a topological direct summand of $G.$ The same assertion is true for the
real numbers In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
$\R$Armacost, David L. The structure of locally compact abelian groups. Monographs and Textbooks in Pure and Applied Mathematics, 68. Marcel Dekker, Inc., New York, 1981. vii+154 pp. MR0637201 (83h:22010)