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On a theorem of E. Helly
Author(s):
Sakaé
Fuchino;
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.
References:
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- E.K. van Douwen, The integers and topology, in: Handbook of Set-Theoretic Topology, edited by K. Kunen and J.E. Vaughan, North-Holland (1984), 111-167. MR 87f:54008
- 3.
- E. Helly, Über lineare Funktionaloperationen, Sitzungsberichte der Naturwiss. Klasse Kais. Akad. Wiss., Wien 121 (1921), 265-295.
- 4.
- H. Rademacher, Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen, Math. Ann. 87 (1922), 112-138.
- 5.
- W. Sierpi\'{n}ski, Remarque sur les suites infinies de fonctions, Fund. Math. XVIII (1932), 110-113.
- 6.
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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 Slaskiego, ul. Bankowa 14, 40 007 Katowice, Poland
Email:
plewik@ux2.math.us.edu.pl
DOI:
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
Copyright of article:
Copyright
1999,
American Mathematical Society
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