Selectors for dense subsets of function spaces

Abstract

Let $USC^*_p(X)$ be the topological space of real upper semicontinuous bounded functions defined on $X$ with the subspace topology of the product topology on ${}^X\mathbb{R}$. $\tilde\Phi^{\uparrow},\tilde\Psi^{\uparrow}$ are the sets of all upper sequentially dense, upper dense or pointwise dense subsets of $USC^*_p(X)$, respectively. We prove several equivalent assertions to the assertion $USC^*_p(X)$ satisfies the selection principles $S_1(\tilde\Phi^{\uparrow},\tilde\Psi^{\uparrow})$, including a condition on the topological space $X$. We prove similar results for the topological space $C^*_p(X)$ of continuous bounded functions. Similar results hold true for the selection principles $S_{fin}(\tilde\Phi^{\uparrow},\tilde\Psi^{\uparrow})$.