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Countable products and sums of lines and circles: their closed subgroups, quotients and duality properties

Published online by Cambridge University Press:  24 October 2008

Ronald Brown ,
Philip J. Higgins  and
Sidney A. Morris
Ronald Brown
Affiliation:
University College of North Wales, Bangor, Gwynedd, Wales
Philip J. Higgins
Affiliation:
King's College, Strand, London, W.C.2, England
Sidney A. Morris
Affiliation:
University of New South Wales, Kensington, N.S.W., Australia
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Extract

It is well-known ((2), Theorem 9·11) that any closed subgroup of Rn is isomorphic (topologically and algebraically) to Ra × Zb, where a, b are suitable non-negative integers. For an infinite product of copies of R, it is also known that any locally compact (hence closed) subgroup is a product of copies R and Z, and that any connected subgroup is a product of copies of R (see (7), (3), respectively). Some information is also given in (3) on closed subgroups of products of copies of R and T, where T = R/Z is the circle group.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 78 , Issue 1 , July 1975 , pp. 19 - 32
Copyright
Copyright © Cambridge Philosophical Society 1975

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References

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