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Continuous isomorphisms from R onto a complete abelian group

Published online by Cambridge University Press:  12 March 2014

Douglas Bridges  and
Matthew Hendtlass
Douglas Bridges
Affiliation:
Department of Mathematics & Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand. E-mail: d.bridges@math.canterbury.ac.nz
Matthew Hendtlass
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds Ls2 9Jt, UK. E-mail: m.hendtlass@gmail.com
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Abstract

This paper provides a Bishop-style constructive analysis of the contrapositive of the statement that a continuous homomorphism of R onto a compact abelian group is periodic. It is shown that, subject to a weak locatedness hypothesis, if G is a complete (metric) abelian group that is the range of a continuous isomorphism from R, then G is noncompact. A special case occurs when G satisfies a certain local path-connectedness condition at 0. A number of results about one-one and injective mappings are proved en route to the main theorem. A Brouwerian example shows that some of our results are the best possible in a constructive framework.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 75 , Issue 3 , September 2010 , pp. 930 - 944
Copyright
Copyright © Association for Symbolic Logic 2010

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