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
$$\backslash begin\; H\_1\backslash times\; H\_2\; \&\backslash longrightarrow\; G\; \backslash \backslash \; (h\_1,h\_2)\; \&\backslash longmapsto\; h\_1\; h\_2\; \backslash end$$
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 ofsubgroup
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 $H\_1,\; \backslash ldots,\; H\_n$ of the map
$$\backslash begin\; \backslash prod^n\_\; H\_i\; \&\backslash longrightarrow\; G\; \backslash \backslash \; (h\_i)\_\; \&\backslash longmapsto\; h\_1\; h\_2\; \backslash cdots\; h\_n\; \backslash end$$
is a topological isomorphism.
If a topological group $G$ is the topological direct sum of the family of subgroups $H\_1,\; \backslash ldots,\; H\_n$ 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 $K\; \backslash leq\; G$ 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 $$0\; \backslash to\; H\backslash stackrel\; G\backslash stackrel\; G/H\backslash to\; 0$$ splits, where $i$ is the natural inclusion and $\backslash pi$ is the natural projection.Examples

Suppose that $G$ is a locally compact abelian group that contains theunit 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 ...

$\backslash mathbb$ as a subgroup. Then $\backslash mathbb$ 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 ...

$\backslash R$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)
See also

* * *References

{{TopologicalVectorSpaces Topological groups Topology