Characterizing function spaces which have the property (B) of Banakh

A topological space Y has the property (B) of Banakh if there is a countable family \Aₙ: n N\ of closed nowhere dense subsets of Y absorbing all compact subsets of Y. In this note we show that the space Cₚ (X) of continuous real-valued functions on a Tychonoff space X with the topology of pointwise convergence, fails to satisfy the property (B) if and only if the space X has the following property (): every sequence of disjoint finite subsets of X has a subsequence with point--finite open expansion. Additionally, we provide an analogous characterization for the compact--open topology on C (X). Finally, we give examples of Tychonoff spaces X whose all bounded subsets are finite, yet X fails to have the property (). This answers a question of Tkachuk.