Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T13:58:14.960Z Has data issue: false hasContentIssue false
  • English
  • Français

The pseudocompactness of [0.1] is equivalent to the uniform continuity theorem

Published online by Cambridge University Press:  12 March 2014

Douglas Bridges  and
Hannes Diener
Douglas Bridges
Affiliation:
Department of Mathematics & Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand. E-mail: [email protected]
Hannes Diener
Affiliation:
Department of Mathematics & Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove constructively that, in order to derive the uniform continuity theorem for pointwise continuous mappings from a compact metric space into a metric space, it is necessary and sufficient to prove any of a number of equivalent conditions, such as that every pointwise continuous mapping of [0, 1] into ℝ is bounded. The proofs are analytic, making no use of, for example, fan-theoretic ideas.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 4 , December 2007 , pp. 1379 - 1384
Copyright
Copyright © Association for Symbolic Logic 2007

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. and Rathjen, M.J., Notes on constructive set theory, Technical Report 40, Institut Mittag-Leffler, Royal Swedish Academy of Sciences, 2001.Google Scholar
[2]Armstrong, M.A., Basic topology. Undergraduate Texts in Mathematics, Springer-Verlag, Heidelberg, 1983.CrossRefGoogle Scholar
[3]Berger, J., Constructive equivalents of the uniform continuity theorem, The Journal of Universal Computer Science, vol. 11 (2005), no. 12, pp. 1878–1883.Google Scholar
[4]Berger, J., The fan theorem and uniform continuity, New computational paradigms (Cooper, S. B., Löwe, B., and Torenvliet, L., editors), Lecture Notes in Computer Science, vol. 3526, Springer-Verlag, 2005, pp. 18–22.CrossRefGoogle Scholar
[5]Berger, J., The logical strength of the uniform continuity theorem, Logical approaches to computational harriers (Beckmann, A., Berger, U., Löwe, B., and Tucker, J., editors), Lecture Notes in Computer Science, vol. 3988, Springer-Verlag, 2006, pp. 35–39.CrossRefGoogle Scholar
[6]Berger, J. and Bridges, D. S., A fan-theoretic equivalent of the antithesis of Specker's theorem, preprint, University of Canterbury, 2006.Google Scholar
[7]Bishop, E. A. and Bridges, D. S., Constructive analysis, Grundlehren der Mathematischen Wissenschaften, vol. 279, Springer-Verlag, Heidelberg, 1985.CrossRefGoogle Scholar
[8]Bridges, D. S., Some notes on continuity in constructive analysis, The Bulletin of the London Mathematical Society, vol. 8 (1976), pp. 179–182.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, New-York, 2006.Google Scholar
[11]Loeb, I., Equivalents of the (weak) fan theorem, Annals of Pure and Applied Logic, vol. 132 (2005), pp. 51–66.CrossRefGoogle Scholar