On property (B) in function spaces

A topological space Z has the property (B) of Banakh if there exists a countable family of closed nowhere dense subsets of Z that absorbs all compact subsets of Z, i.e., each compact subset of Z is contained in some member of this family. During the talk, we will demonstrate that the space Cp(X) of continuous real-valued functions on a Tychonoff space X, equipped with the topology of pointwise convergence, fails to possess property (B) if and only if the space X satisfies the following property (\kappa): every sequence of pairwise disjoint finite subsets of X has an infinite subsequence with a point-finite open expansion. Additionally, time permitting, we will provide an analogous characterization for the compact-open topology on C(X).
The results come from the still ongoing work with Mikołaj Krupski.