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