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On the structure of non-dentable subsets of $C(\omega ^{\omega ^{{k}}} )$

Volume 203 / 2011

Pericles D. Pavlakos, Minos Petrakis Studia Mathematica 203 (2011), 205-222 MSC: 46B20, 46B22. DOI: 10.4064/sm203-3-1

Abstract

It is shown that there is no closed convex bounded non-dentable subset $K$ of $C(\omega ^{\omega ^{k}})$ such that on subsets of $K$ the PCP and the RNP are equivalent properties. Then applying the Schachermayer–Rosenthal theorem, we conclude that every non-dentable $K$ contains a non-dentable subset $L$ so that on $L$ the weak topology coincides with the norm topology. It follows from known results that the RNP and the KMP are equivalent on subsets of $C(\omega ^{\omega ^{k}})$.

Authors

  • Pericles D. PavlakosDepartment of Sciences, Section of Mathematics
    Technical University of Crete
    73100 Chania, Greece
    e-mail
  • Minos PetrakisDepartment of Sciences, Section of Mathematics
    Technical University of Crete
    73100 Chania, Greece
    e-mail

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