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BROUWER’S FAN THEOREM AND CONVEXITY

Published online by Cambridge University Press:  21 December 2018

JOSEF BERGER
Affiliation:
MATHEMATISCHES INSTITUT LUDWIG-MAXIMILIANS-UNIVERSITÄT MÜNCHEN THERESIENSTRASSE 39 80333 MÜNCHEN, GERMANYE-mail:jberger@math.lmu.de
GREGOR SVINDLAND
Affiliation:
MATHEMATISCHES INSTITUT LUDWIG-MAXIMILIANS-UNIVERSITÄT MÜNCHEN THERESIENSTRASSE 39 80333 MÜNCHEN, GERMANYE-mail:svindla@math.lmu.de

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.

Type
Articles
Copyright
Copyright © The Association for Symbolic Logic 2018 

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References

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