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BROUWER’S FAN THEOREM AND CONVEXITY

  • JOSEF BERGER (a1) and GREGOR SVINDLAND (a2)

Abstract

In the framework of Bishop’s constructive mathematics we introduce co-convexity as a property of subsets B of ${\left\{ {0,1} \right\}^{\rm{*}}}$ , the set of finite binary sequences, and prove that co-convex bars are uniform. Moreover, we establish a canonical correspondence between detachable subsets B of ${\left\{ {0,1} \right\}^{\rm{*}}}$ and uniformly continuous functions f defined on the unit interval such that B is a bar if and only if the corresponding function f is positive-valued, B is a uniform bar if and only if f has positive infimum, and B is co-convex if and only if f satisfies a weak convexity condition.

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[1]Berger, J. and Bridges, D., A fan-theoretic equivalent of the antithesis of Specker’s theorem. Mathematicae, vol. 18 (2007), pp. 195202.
[2]Berger, J. and Ishihara, H., Brouwer’s fan theorem and unique existence in constructive analysis. Mathematical Logic Quarterly, vol. 51 (2005), pp. 360364.
[3]Berger, J. and Svindland, G., Convexity and constructive infima. Archive for Mathematical Logic, vol. 55 (2016), pp. 873881.
[4]Bishop, E. and Bridges, D., Constructive Analysis, Springer-Verlag, Heidelberg, 1985.
[5]Bridges, D. and Richman, F., Varieties of Constructive Mathematics, London Mathematical Society Lecture Notes, vol. 97, Cambridge University Press, New York, 1987.
[6]Bridges, D. and Vita, L. S., Techniques of Constructive Analysis, Universitext, Springer-Verlag, New York, 2006.
[7]Julian, W. and Richman, F., A uniformly continuous function on $[0,1]$ that is everywhere different from its infimum. Pacific Journal of Mathematics, vol. 111 (1984), pp. 333340.

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BROUWER’S FAN THEOREM AND CONVEXITY

  • JOSEF BERGER (a1) and GREGOR SVINDLAND (a2)

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