Direct sum of topological groups

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

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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

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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

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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

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References

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  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)