In
mathematics, a
topological group is called the topological direct sum of two
subgroup
In group theory, a branch of mathematics, given a 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, ''H'' is a subgrou ...
s
and
if the map
is a topological isomorphism, meaning that it is a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
and a
group isomorphism.
Definition
More generally,
is called the direct sum of a finite set of
subgroup
In group theory, a branch of mathematics, given a 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, ''H'' is a subgrou ...
s
of the map
is a topological isomorphism.
If a topological group
is the topological direct sum of the family of subgroups
then in particular, as an abstract group (without topology) it is also the
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
(in the usual way) of the family
Topological direct summands
Given a topological group
we say that a subgroup
is a topological direct summand of
(or that splits topologically from
) if and only if there exist another subgroup
such that
is the direct sum of the subgroups
and
A the subgroup
is a topological direct summand if and only if the
extension of topological groups
splits, where
is the natural inclusion and
is the natural projection.
Examples
Suppose that
is a
locally compact abelian group that contains the
unit circle as a subgroup. Then
is a topological direct summand of
The same assertion is true for the
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
[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