Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-19T05:40:49.166Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Matthew Hendtlass
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds Ls2 9Jt, UK. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Aczel, P. H. G. and Rathjen, M., Notes on constructive set theory, Report 40, Institut Mittag-Leffler, Royal Swedish Academy of Sciences, 2001.Google Scholar
[2]Berger, J. and Bridges, D. S., A fan-theoretic equivalent of the antithesis of Specker's theorem, Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae, New Series, vol. 18 (2007), no. 2, pp. 195–202.Google Scholar
[3]Bishop, E. A., Foundations of constructive analysis, McGraw-Hill, New York, 1967.Google Scholar
[4]Bishop, E. A. and Bridges, D. S., Constructive analysis, Grundlehren der Mathematischen Wissenschaften, vol. 279, Springer, 1985.CrossRefGoogle Scholar
[5]Bridges, D. S., A constructive morse theory of sets, Mathematical logic and its applications (Skordev, D., editor), Plenum Press, New York, 1987, pp. 61–79.CrossRefGoogle Scholar
[6]Bridges, D. S., Omniscience, sequential compactness, and the anti-Specker property, preprint, University of Canterbury, 2008.Google Scholar
[7]Bridges, D. S. and Hendtlass, M., Continuous homomorphisms from R onto a compact abelian group are periodic, Mathematical Logic Quarterly, to appear.Google Scholar
[8]Bridges, D. S. and Ishihara, H., Linear mappings are fairly well-behaved, Archiv der Mathematik, vol. 54 (1990), pp. 558–562.CrossRefGoogle Scholar
[9]Bridges, D. S. and Richman, F., Varieties of constructive mathematics, London Mathematical Society Lecture Notes, vol. 97, Cambridge University Press, 1987.CrossRefGoogle Scholar
[10]Bridges, D. S. and Vîţӑ, L. S., Techniques of constructive analysis, Universitext, Springer, 2006.Google Scholar
[11]Dummett, M. A. E., Elements of intuitionism, second ed., Oxford Logic Guides, vol. 39, Clarendon Press, Oxford, 2000.CrossRefGoogle Scholar
[12]Friedman, H. M., Set theoretic foundations for constructive analysis, Annals of Mathematics, vol. 105 (1977), no. 1. pp. 1–28.CrossRefGoogle Scholar
[13]Irwin, M. C., Smooth dynamical systems, World Scientific, Singapore, 2001.CrossRefGoogle Scholar
[14]Ishihara, H., An omniscience principle, the König lemma and the Hahn–Banach theorem, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 36 (1990), pp. 237–240.CrossRefGoogle Scholar
[15]Ishihara, H., Continuity properties in constructive mathematics, this Journal, vol. 57 (1992), no. 2, pp. 557–565.Google Scholar
[16]Ishihara, H. and Schuster, P. M., A continuity principle, a version of Baire's theorem and a boundedness principle, this Journal, (to appear).Google Scholar
[17]Lietz, P., From constructive mathematics to computable analysis via the realizability interpretation, Ph.D. thesis, Technische Universität, Darmstadt, 2004.Google Scholar
[18]Myhill, J., Constructive set theory, this Journal, vol. 40 (1975), no. 3, pp. 347–382.Google Scholar
[19]Richman, F., A new constructive proof of Baire's theorem(s), 2008, preprint, Florida Atlantic University.Google Scholar