Chains of functions in C(K)-spaces

Abstract

The Bishop property (+), 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 (+): 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 (+) was first introduced. We show that if D is a class of compact spaces that is preserved when taking closed subspaces and Hausdorff quotients, and which contains no nonmetrizable linearly ordered space, then every member of D has (+). Examples of such classes include all K for which C(K)is Lindelöf 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 (+) does not form a right ideal in (C(K)). Some results regarding local connectedness are also presented.