Selectors for dense subsets of function spaces

Let USCp ⋆(X) be the topological space of real upper semicontinuous bounded functions defined on X with the subspace topology of the product topology on RX. Φ˜↑,Ψ˜↑ are the sets of all upper sequentially dense, upper dense or pointwise dense subsets of USCp ⋆(X), respectively. We prove several equivalent assertions to that USCp ⋆(X) satisfies the selection principles S1(Φ˜↑,Ψ˜↑), including a condition on the topological space X. We prove similar results for the topological space Cp ⋆(X) of continuous bounded functions. Similar results hold true for the selection principles Sfin(Φ˜↑,Ψ˜↑). © 2019 Elsevier B.V