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σ-fragmentable Banach spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  26 February 2010

J. E. Jayne ,
I. Namioka  and
C. A. Rogers
J. E. Jayne
Affiliation:
Department of Mathematics, University College London, Gower Street, London. WC1E 6BT
I. Namioka
Affiliation:
Department of Mathematics, University of Washington, Seattle, WA 98195, U.S.A.
C. A. Rogers
Affiliation:
Department of Mathematics, University College London, Gower Street, London. WC1E 6BT
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§1. Introduction. Let X be a Hausdorff space and let ρ be a metric, not necessarily related to the topology of X. The space X is said to be fragmented by the metric ρ if each non-empty set in X has non-empty relatively open subsets of arbitrarily small ρ-diameter. The space X is said to be a σ-fragmented by the metric ρ if, for each ε>0, it is possible to write

where each set Xi, i≥1, has the property that each non-empty subset of Xi, has a non-empty relatively open subset of ρ-diameter less than ε. If is any family of subsets of X, we say that X is σ-fragmented by the metric ρ, using sets from, if, for each ε>0, the sets Xi, i ≥ 1, in (1.1) can be taken from

MSC classification

Secondary: 46B26: Nonseparable Banach spaces
Type
Research Article
Information
Mathematika , Volume 39 , Issue 1 , June 1992 , pp. 161 - 188
Copyright
Copyright © University College London 1992

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