Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. B. Balcar, P. Simon, Disjoint refinement, in: Handbook of Boolean Algebras, edited by J. D. Monk with R. Bonnet, Elsevier Science Publishers B.V. (1989), 333-386.
  • 2. Eric K. van Douwen, The integers and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR 776622
  • 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. A. Tychonoff, Über einen Funktionenraum, Math. Ann. 111 (1935), 762-766.
  • 7. J.E. Vaughan, Small uncountable cardinals and topology, in: Open Problems in Topology, edited by J. van Mill and G.M. Reed, North-Holland (1990), 195-218. CMP 91:03
  • 8. Peter Vojtáš, More on set-theoretic characteristics of summability of sequences by regular (Toeplitz) matrices, Comment. Math. Univ. Carolin. 29 (1988), no. 1, 97–102. MR 937553

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 26A03, 06A05, 03E10, 03E35

Retrieve articles in all journals with MSC (1991): 26A03, 06A05, 03E10, 03E35

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

Szymon Plewik
Affiliation: Instytut Matematyki Uniwersytetu Ślaskiego, ul. Bankowa 14, 40 007 Katowice, Poland

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