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 A ¯ . Every space with the Collins–Roscoe property is monotonically monolithic. We show that for any uncountable discrete space D, the space C p ( β D ) does not have the Collins–Roscoe property; since C p ( β D ) 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 n w ( X ) ⩽ ω 1 then C p ( X ) has the Collins–Roscoe property; this implies that C p ( 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 C p ( X ) are Lindelöf Σ-spaces, then the iterated function space C p , n ( X ) has the Collins–Roscoe property for every n ∈ ω .