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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On a theorem of E. Helly


Authors: Sakaé Fuchino and Szymon Plewik
Journal: Proc. Amer. Math. Soc. 127 (1999), 491-497
MSC (1991): Primary 26A03, 06A05, 03E10, 03E35
MathSciNet review: 1468190
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Abstract: E. Helly's theorem asserts that any bounded sequence of monotone real functions contains a pointwise convergent subsequence. We reprove this theorem in a generalized version in terms of monotone functions on linearly ordered sets. We show that the cardinal number responsible for this generalization is exactly the splitting number. We also show that a positive answer to a problem of S. Saks is obtained under the assumption of the splitting number being strictly greater than the first uncountable cardinal.


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Additional Information

Sakaé Fuchino
Affiliation: Institut für Mathematik II, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany
Address at time of publication: Department of Computer Sciences, Kitami Institute of Technology, Kitami, Hokkaido 090 Japan
Email: fuchino@math.fu-berlin.de, fuchino@math.cs.kitami-it.ac.jp

Szymon Plewik
Affiliation: Instytut Matematyki Uniwersytetu Ślaskiego, ul. Bankowa 14, 40 007 Katowice, Poland
Email: plewik@ux2.math.us.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04540-2
PII: S 0002-9939(99)04540-2
Keywords: Helly's theorem, splitting number, Saks' problem
Received by editor(s): August 8, 1996
Received by editor(s) in revised form: May 26, 1997
Communicated by: Andreas R. Blass
Article copyright: © Copyright 1999 American Mathematical Society