The Rothberger property on Cp(X,2)

In this paper we analyze the Rothberger property on Cp(X,2). A space X is said to have the Rothberger property (or simply X is Rothberger) if for every sequence 〈Un:n∈ω〉 of open covers of X, there exists Un∈Un for each n∈ω such that X=⋃n∈ωUn. We show the following: (1) if Cp(X,2) is Rothberger, then X is pseudocompact; (2) for every pseudocompact Sokolov space X with t⁎(X)≤ω, Cp(X,2) is Rothberger; and (3) assuming CH (the continuum hypothesis) there is a maximal almost disjoint family A for which the space Cp(Ψ(A),2) is Rothberger. Moreover, we characterize the Rothberger property on Cp(L,2) when L is a GO-space.