On a theorem of E. Helly

Fuchino, Sakaé; Plewik, Szymon

doi:10.1090/S0002-9939-99-04540-2

On a theorem of E. Helly
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by Sakaé Fuchino and Szymon Plewik PDF

Proc. Amer. Math. Soc. 127 (1999), 491-497 Request permission

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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