Jump to content

Direct sum of topological groups

From Wikipedia, the free encyclopedia

In mathematics, a topological group is called the topological direct sum[1] of two subgroups and if the map is a topological isomorphism, meaning that it is a homeomorphism and a group isomorphism.

Definition

[edit]

More generally, is called the direct sum of a finite set of subgroups 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 (in the usual way) of the family

Topological direct summands

[edit]

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

[edit]

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 [2]

See also

[edit]

References

[edit]
  1. ^ E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I, second edition, Grundlehren der Mathematischen Wissenschaften, 115, Springer, Berlin, 1979. MR0551496 (81k:43001)
  2. ^ 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. ISBN 0-8247-1507-1 MR0637201 (83h:22010)