Étude constructive de problèmes de topologie pour les réels irrationnels

Mathematical Logic Quarterly 45 (2):257-288 (1999)
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Abstract

We study in a constructive manner some problems of topology related to the set Irr of irrational reals. The constructive approach requires a strong notion of an irrational number; constructively, a real number is irrational if it is clearly different from any rational number. We show that the set Irr is one-to-one with the set Dfc of infinite developments in continued fraction . We define two extensions of Irr, one, called Dfc1, is the set of dfc of rationals and irrationals preserving for each rational one dfc, the other, called Dfc2, is the set of dfc of rationals and irrationals preserving for each rational its two dfc. We introduce six natural distances over Irr wich we denote by dfc0, dfc1, dfc2, d, dmir and dcut. We show that only the four distances dfco, dfc1, d and dmir among the six make Irr a complete metric space. The last distances define in Irr the same topology in a constructive sens. We study further the set Dfc1 in which we show that the irrationals constitute a closed subset. Finally, we make a particular study of the completion Dfc2 of Dfc for the two equivalent metrics dfc2 and dcut

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References found in this work

Foundations of Constructive Analysis.John Myhill - 1972 - Journal of Symbolic Logic 37 (4):744-747.
Varieties of constructive mathematics.D. S. Bridges - 1987 - New York: Cambridge University Press. Edited by Fred Richman.
Constructive Analysis.Errett Bishop & Douglas Bridges - 1987 - Journal of Symbolic Logic 52 (4):1047-1048.
Constructive Functional Analysis.D. S. Bridges & Peter Zahn - 1982 - Journal of Symbolic Logic 47 (3):703-705.
Foundations of Constructive Mathematics.Michael J. Beeson - 1987 - Studia Logica 46 (4):398-399.

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