Skip to main content Accessibility help
×
Home
Hostname: page-component-cf9d5c678-9z9qw Total loading time: 0.159 Render date: 2021-08-01T08:41:55.172Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

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

Published online by Cambridge University Press:  12 March 2014

Douglas Bridges
Affiliation:
Department of Mathematics & Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand. E-mail: d.bridges@math.canterbury.ac.nz
Hannes Diener
Affiliation:
Department of Mathematics & Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand. E-mail: H.Diener@math.canterbury.ac.nz

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
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

[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. 18781883.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. 1822.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. 3539.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. 179182.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. 5166.CrossRefGoogle Scholar
8
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

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

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

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

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

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

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *