The Collins–Roscoe property and its applications in the theory of function spaces

A space X has the Collins–Roscoe property if we can assign, to each x∈X, a family G(x) of subsets of X in such a way that for every set A⊂X, the family ⋃{G(a):a∈A} contains an external network of . Every space with the Collins–Roscoe property is monotonically monolithic. We show that for any uncountable discrete space D, the space Cp(βD) does not have the Collins–Roscoe property; since Cp(βD) is monotonically monolithic, this proves that monotone monolithity does not imply the Collins–Roscoe property and provides an answer to two questions of Gruenhage. However, if X is a Lindelöf Σ-space with nw(X)⩽ω1 then Cp(X) has the Collins–Roscoe property; this implies that Cp(X) is metalindelöf and constitutes a generalization of an analogous theorem of Dow, Junnila and Pelant proved for a compact space X. We also establish that if X and Cp(X) are Lindelöf Σ-spaces, then the iterated function space Cp,n(X) has the Collins–Roscoe property for every n∈ω.