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ON $\omega$-INDEPENDENCE AND THE KUNEN–SHELAH PROPERTY

Published online by Cambridge University Press:  17 June 2002

A. S. Granero ,
M. Jiménez-Sevilla  and
J. P. Moreno
A. S. Granero
Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain (granero@mat.ucm.es; marjim@mat.ucm.es)
M. Jiménez-Sevilla
Affiliation:
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain (granero@mat.ucm.es; marjim@mat.ucm.es)
J. P. Moreno
Affiliation:
Departamento de Matemáticas, Universidad Autonoma de Madrid, 28049 Madrid, Spain (josepedro.moreno@uam.es)
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Abstract

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We prove that spaces with an uncountable $\omega$-independent family fail the Kunen–Shelah property. Actually, if $\{x_i\}_{i\in I}$ is an uncountable $\omega$-independent family, there exists an uncountable subset $J\subset I$ such that $x_j\notin\overline{\co}(\{x_i\}_{i\in J\setminus\{j\}})$ for every $j\in J$. This improves a previous result due to Sersouri, namely that every uncountable $\omega$-independent family contains a convex right-separated subfamily.

AMS 2000 Mathematics subject classification: Primary 46B20; 46B26

Keywords

$\omega$-independencenon-separable Banach spacesKunen–Shelah property

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 45 , Issue 2 , June 2002 , pp. 391 - 395
Copyright
Copyright © Edinburgh Mathematical Society 2002