Abstract of a paper by Tomasz Kania and Richard J. Smith
This is an announcement for the paper "A note on the Bishop property in compact spaces" by Tomasz Kania and Richard J. Smith. Abstract: We answer two questions concerning the Bishop property ($\symbishop$), introduced recently by K.P. Hart, T. Kochanek and the first-named author. There are two versions of ($\symbishop$): one applies to linear operators and the other to compact Hausdorff spaces. 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. Archive classification: math.GN math.FA Submitted from: t.kania@lancaster.ac.uk The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1310.4035 or http://arXiv.org/abs/1310.4035
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