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Continuity properties in constructive mathematics

Published online by Cambridge University Press:  12 March 2014

Hajime Ishihara
Hajime Ishihara*
Affiliation:
Faculty of Integrated Arts and Sciences, Hiroshima University, Hiroshima 730, Japan, E-mail: ishihara@mis.hiroshima-u.ac.jp
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Abstract

The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We deal with principles which are equivalent to the statements “every mapping is sequentially nondiscontinuous”, “every sequentially nondiscontinuous mapping is sequentially continuous”, and “every sequentially continuous mapping is continuous”. As corollaries, we show that every mapping of a complete separable space is continuous in constructive recursive mathematics (the Kreisel-Lacombe-Schoenfield-Tsejtin theorem) and in intuitionism.

Keywords

Continuitysequential continuitysequential nondiscontinuityKreisel-Lacombe-Shoenfield-Tsejtin theoremconstructive

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 57 , Issue 2 , June 1992, pp. 557 - 565
Copyright
Copyright © Association for Symbolic Logic 1992

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References

REFERENCES

[1]Beeson, M., The nonderivability in intuitionistic formal system of theorems on the continuity of effective operations, this Journal, vol. 40 (1983), pp. 321–346.Google Scholar
[2]Beeson, M., Foundations of constructive mathematics, Springer-Verlag, Berlin, 1985.CrossRefGoogle Scholar
[3]Bishop, E., Foundations of constructive analysis, McGraw-Hill, New York, 1967.Google Scholar
[4]Bishop, E. and Bridges, D., Constructive analysis, Springer-Verlag, Berlin, 1985.CrossRefGoogle Scholar
[5]Bridges, D. and Richman, F., Varieties of constructive mathematics, London Mathematical Society Lecture Note Series, vol. 97, Cambridge University Press, Cambridge, 1987.CrossRefGoogle Scholar
[6]Ishihara, H., Continuity and nondiscontinuity in constructive mathematics, this Journal, vol. 56 (1991), pp. 1349–1354.Google Scholar
[7]Kreisel, G., Lacombe, D., and Shoenfield, J., Fonctionnelles récursivement définissables et fonctionnelles récursives, Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, vol. 245 (1957), pp. 399–402.Google Scholar
[8]Kushner, B. A., Lectures on constructive mathematical analysis, “Nauka”, Moscow, 1973; English translation, American Mathematical Society, Providence, Rhode Island, 1985.Google Scholar
[9]Mandelkern, M., Constructive complete finite sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 34 (1988), pp. 97–103.CrossRefGoogle Scholar
[10]Orevkov, V. P.,Equivalence of two definitions of continuity, Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR (LOMI), vol. 20 (1971), pp. 145–159; English translation, Journal of Soviet Mathematics, vol. 1 (1973), pp. 92–99.Google Scholar
[11]Richman, F., Church's thesis without tears, this Journal, vol. 48 (1983), pp. 797–803.Google Scholar
[12]Troelstra, A. S., Intuitionistic continuity, Nieuw Archief voor Wiskunde, ser. 3, vol. 15 (1967), pp. 2–6.Google Scholar
[13]Troelstra, A. S. and van Dalen, D., Constructivism in mathematics, Vol. 1, North-Holland, Amsterdam, 1988.Google Scholar
[14]Troelstra, A. S. and van Dalen, D., Constructivism in mathematics, Vol. 2, North-Holland, Amsterdam, 1988.Google Scholar
[15]Tsejtin, G. S., Algorithmic operators in constructive complete metric spaces, Doklady Akademii Nauk SSSR, vol. 128 (1959), pp. 49–52, (Russian)Google Scholar