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Compactness and Continuity, Constructively Revisited

  • Douglas Bridges
  • Hajime Ishihara
  • Peter Schuster
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2471)

Abstract

In this paper, the relationships between various classical compactness properties, including the constructively acceptable one of total boundedness and completeness, are examined using intuitionistic logic. For instance, although every metric space clearly is totally bounded whenever it possesses the Heine-Borel property that every open cover admits of a finite subcover, we show that one cannot expect a constructive proof that any such space is also complete. Even the Bolzano-Weierstraβ principle, that every sequence in a compact metric space has a convergent subsequence, is brought under our scrutiny; although that principle is essentially nonconstructive, we produce a reasonable, classically equivalent modification of it that is constructively valid. To this end, we require each sequence under consideration to satisfy uniformly a classically trivial approximate pigeonhole principle—that if infinitely many elements of the sequence are close to a finite set of points, then infinitely many of those elements are close to one of these points—whose constructive failure for arbitrary sequences is then detected as the obstacle to any constructive relevance of the traditional Bolzano-Weierstraβ principle.

2000 MSC (AMS) 03F60 26E40 54E45 

Keywords

Compact Metric Spaces Uniform Continuity Constructive Analysis 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Douglas Bridges
    • 1
  • Hajime Ishihara
    • 2
  • Peter Schuster
    • 3
  1. 1.Department of Mathematics & StatisticsUniversity of CanterburyChristchurchNew Zealand
  2. 2.School of Information Science, Japan Advanced Institute of Science and TechnologyTatsunokuchi, IshikawaJapan
  3. 3.Mathematisches InstitutLudwig-Maximilians-Universität MünchenMünchenGermany

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