Selectors for dense subsets of function spaces

Abstract

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˜↑).