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Radon spaces which are not \(\sigma\)-fragmentable. (English) Zbl 0853.54025

The concept of \(\sigma\)-fragmentability was introduced by J. Jayne, I. Namioka, and C. A. Rogers [J. Lond. Math. Soc., III. Ser. 66, 651-672 (1993; Zbl 0793.54026)]. J. Jayne raised the question in 1994 whether, for a Banach space endowed with the weak topology, \(\sigma\)-fragmented and Radon are the same provided that the space does not contain a relatively discrete subset of real-valued measurable cardinality. In this paper, a negative answer is given to this question by considering spaces of continuous functions on tree spaces. It is shown that these spaces are Radon with the above restriction on relatively discrete subsets. Then some tree spaces are described for which the weak topology of the corresponding Banach space of continuous functions turns out to be not \(\sigma\)-fragmented by any metric, or at least, by the supremum norm.

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)

Keywords:

tree spaces

Citations:

Zbl 0793.54026
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