Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Chains of functions in $C(K)$-spaces   The Bishop property ($\symbishop$), introduced recently by K.P. Hart, T.
Kochanek and the first-named author, was motivated by Pe{\l}czy{\'n}ski'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.

21
The categories of this paper are: adap-org
23
The categories of this paper are: solv-int
25
The categories of this paper are: patt-sol
28
The categories of this paper are: alg-geom
19
The categories of this paper are: eess
4
The categories of this paper are: cond-mat
35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex
7
The categories of this paper are: hep-th
29
The categories of this paper are: funct-an
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ