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)


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 


Compact Metric Spaces Uniform Continuity Constructive Analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Michael J. Beeson, Foundations of Constructive Mathematics, Ergebn. Math. Grenzgeb. Math. (3) 6, Springer, Heidelberg, 1985.Google Scholar
  2. [2]
    Errett Bishop, Foundations of Constructive Analysis, McGraw-Hill, New York, 1967.zbMATHGoogle Scholar
  3. [3]
    Errett Bishop and Douglas Bridges, Constructive Analysis, Grundlehr. math. Wiss. 279, Springer, Heidelberg, 1985.Google Scholar
  4. [4]
    Douglas S. Bridges, Foundations of Real and Abstract Analysis, Graduate Texts Math. 174, Springer, New York, 1998.Google Scholar
  5. [5]
    Douglas S. Bridges, ‘Constructive mathematics: a foundation for computable analysis’, Theoret. Comput. Sci. 219, 95–109, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Douglas S. Bridges, ‘Prime and maximal ideals in constructive ring theory’, Communic. Algebra 29, 2787–2803, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Douglas Bridges, Hajime Ishihara, and Peter Schuster, ‘Sequential compactness in constructive analysis’, Osterreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 208, 159–163, 1999.zbMATHMathSciNetGoogle Scholar
  8. [8]
    Douglas Bridges and Ayan Mahalanobis, ‘Bounded variation implies regulated: a constructive proof’, J. Symb. Logic 66, 1695–1700, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Douglas Bridges and Fred Richman, Varieties of Constructive Mathematics, London Math. Soc. Lect. Notes Math. 97, Cambridge University Press, 1987.Google Scholar
  10. [10]
    Douglas Bridges, Fred Richman, and Peter Schuster, ‘A weak countable choice principle’, Proc. Amer. Math. Soc. 128(9), 2749–2752, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Douglas Bridges, Fred Richman, and Wang Yuchuan, ‘Sets, complements and boundaries’, Proc. Koninklijke Nederlandse Akad. Wetenschappen (Indag. Math.,N.S.) 7(4), 425–445, 1996.zbMATHMathSciNetGoogle Scholar
  12. [12]
    Nicolas D. Goodman and John Myhill, ‘Choice implies excluded middle’, Zeit. Math. Logik Grundlag. Math. 24, 461, 1978.CrossRefMathSciNetGoogle Scholar
  13. [13]
    Hajime Ishihara, ‘An omniscience principle, the König lemma and the Hahn-Banach theorem’. Zeit. Math. Logik Grundlag. Math. 36, 237–240, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Hajime Ishihara and Peter Schuster, ‘Constructive compactness continued’. Preprint, University of Munich, 2001.Google Scholar
  15. [15]
    Fred Richman, ‘Constructive mathematics without choice’. In: Peter Schuster, Ulrich Berger, and Horst Osswald, eds., Reuniting the Antipodes. Constructive and Nonstandard Views of the Continuum. Proc. 1999 Venice Symposion. Synthese Library 306, 199–205. Kluwer, Dordrecht, 2001.Google Scholar
  16. [16]
    Peter M. Schuster, ‘Unique existence, approximate solutions, and countable choice’, Theoret. Comput. Sci., to appear.Google Scholar
  17. [17]
    Rudolf Taschner, Lehrgang der konstruktiven Mathematik (three volumes), Manz and Hölder-Pichler-Tempsky, Wien, 1993, 1994, 1995.Google Scholar
  18. [18]
    Anne S. Troelstra and Dirk van Dalen, Constructivism in Mathematics (two volumes), North-Holland, Amsterdam, 1988.Google Scholar

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

Personalised recommendations