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Strong and Uniform Continuity – the Uniform Space Case

Published online by Cambridge University Press:  01 February 2010

Douglas Bridges  and
Luminiţa Vîţă
Douglas Bridges
Affiliation:
Department of Mathematics & Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand [email protected], http://www.math.canterbury.ac.nz/~mathdsb
Luminiţa Vîţă
Affiliation:
Department of Mathematics & Statistics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand [email protected], http://www.math.canterbury.ac.nz/~mathlsv

Abstract

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It is proved, within the constructive theory of apartness spaces, that a strongly continuous mapping from a totally bounded uniform space with a countable base of entourages to a uniform space is uniformly continuous. This lifts a result of Ishihara and Schuster from metric to uniform apartness spaces. The paper is part of a systematic development of computable topology using apartness as the fundamental notion.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 6 , 2003 , pp. 326 - 334
Copyright
Copyright © London Mathematical Society 2003

References

1Aberth, O., Computable analysis (McGraw-Hill, New York, 1980). 326Google Scholar
2Beeson, M.J., Foundations of constructive mathematics, Ergeb. Math. Grenzgeb. (3) 6 (Springer, Berlin, 1985). 327CrossRefGoogle Scholar
3Bishop, E.A., Foundations of constructive analysis (McGraw-Hill, New York, 1967). 327Google Scholar
4Bishop, E.A. and Bridges, D.S., Constructive analysis, Grundlehren Math. Wiss. 279 (Springer, Heidelberg, 1985).Google Scholar
5Bourbaki, N., General topology (Springer, Heidelberg, 1989) Chapters 5–10.CrossRefGoogle Scholar
6Bridges, D.S. and Richman, F., Varieties of constructive mathematics, London Math. Soc. Lecture Note Ser. 95 (Cambridge Univ. Press, London, 1987).CrossRefGoogle Scholar
7Bridges, D.S. and Vîţă, L.S., ‘Cauchy nets in the constructive theory of apartness spaces’, Sci. Math. Jpn. 56 (2002) 123–132.Google Scholar
8, D.S. Bridges and Vîţă, L.S., Apartness spaces as a framework for constructive topology’, Ann. Pure Appl. Logic 119 (2003) 61–83.Google Scholar
9Bridges, D.S. and Vîţă, L.S., ‘A proof-technique in uniform space theory’, J. Symbolic Logic 68 (2003) 95–802.CrossRefGoogle Scholar
10Bridges, D.S. and Vîţă, L.S., More on Cauchy nets in apartness spaces’, Sci. Math. Jpn., to appear.Google Scholar
11Bridges, D.S., Ishihara, H., Schuster, P.M. and Vîţă, L.S., ‘Strong continuity implies uniform sequential continuity’, preprint, University of Canterbury, Christchurch, New Zealand, 2000.Google Scholar
12Bridges, D.S., Schuster, P.M. and Vîţă, L.S., ‘Apartness, topology, and uniformity: a constructive view’, Math. Logic Quarterly 48 (2002) special issue: Computability and complexity in analysis (Proc. Dagstuhl Seminar 01461, 11–16 November 2001) Suppl. 1, 16–28.3.0.CO;2-7>CrossRefGoogle Scholar
13Ishihara, H. and Schuster, P.M., ‘A constructive uniform continuity theorem’, Quart. J. Math 53 (2002) 185–193.CrossRefGoogle Scholar
14Kushner, B.A., Lectures on constructive mathematical analysis, Transl. Math. Monogr. 60 (Amer. Math. Soc, Providence, RI, 1985).Google Scholar
15Naimpally, S.A. and Warrack, B.D., Proximity spaces, Cambridge Tracts in Math, and Math. Phys. 59 (Cambridge Univ. Press, London, 1970).Google Scholar
16Schuster, P.M., Vîţă, L.S. and Bridges, D.S., ‘Apartness as a relation between subsets’, Combinatorics, computability and logic (Proceedings of DMTCS'01, Constanţa, Romania, 2–6 July 2001), DMTCS Series 17 (ed. Calude, C.S., Dinneen, M.J. and Sburlan, S., Springer, London, 2001) 203–214.Google Scholar
17Troelstra, A.S. and Van Dalen, D., Constructivism in mathematics: an introduction, 2 volumes (North Holland, Amsterdam, 1988).Google Scholar
18Vîţă, L.S. and Bridges, D.S., ‘A constructive theory of point-set nearness’, Theoret. Comput. Sci. 305(2003) special issue: Topology in computer science: constructivity; asymmetry and partiality; digitization (Proc. Dagstuhl Seminar 00231,4-9 June 2000, ed. Kopperman, R., Smyth, M. and Spreen, D.)473–489.Google Scholar
19Weihrauch, K., Computable analysis, EATCS Texts in Theoret. Comput. Sci. (Springer, Heidelberg, 2000).CrossRefGoogle Scholar