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    Rights statement: http://journals.cambridge.org/action/displayJournal?jid=JAZ The final, definitive version of this article has been published in the Journal, Journal of the Australian Mathematical Society, 99 (3), pp 350-363 2015, © 2015 Cambridge University Press.

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Chains of functions in C(K)-spaces

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  • Tomasz Kania
  • Richard J. Smith
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<mark>Journal publication date</mark>12/2015
<mark>Journal</mark>Journal of the Australian Mathematical Society
Issue number3
Volume99
Number of pages14
Pages (from-to)350-363
Publication StatusPublished
Early online date1/09/15
<mark>Original language</mark>English

Abstract

The Bishop property ($\symbishop$), introduced recently by K.P. Hart, T. Kochanek and the first-named author, was motivated by Pełczyński's classical work on weakly compact operators on $C(K)$-spaces. This property asserts that certain chains of functions in said spaces, with respect to a particular partial ordering, must be countable. There are two versions of ($\symbishop$): one applies to linear operators on $C(K)$-spaces and the other to the compact Hausdorff spaces themselves. We answer two questions that arose after ($\symbishop$) was first introduced. We show that if $\mathscr{D}$ is a class of compact spaces that is preserved when taking closed subspaces and Hausdorff quotients, and which contains no non-metrizable linearly ordered space, then every member of $\mathscr{D}$ has ($\symbishop$). Examples of such classes include all $K$ for which $C(K)$ is Lindel\"of in the topology of pointwise convergence (for instance, all Corson compact spaces) and the class of Gruenhage compact spaces. We also show that the set of operators on a $C(K)$-space satisfying ($\symbishop$) does not form a right ideal in $\mathscr{B}(C(K))$. Some results regarding local connectedness are also presented.

Bibliographic note

http://journals.cambridge.org/action/displayJournal?jid=JAZ The final, definitive version of this article has been published in the Journal, Journal of the Australian Mathematical Society, 99 (3), pp 350-363 2015, © 2015 Cambridge University Press.