Selectors for dense subsets of function spaces

Abstract

Let $US{C}_p^{\ast }(X)$ 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 $US{C}_p^{\ast }(X)$, respectively. We prove several equivalent assertions to the assertion $US{C}_p^{\ast }(X)$ 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 $C_p^{\ast }(X)$ of continuous bounded functions. Similar results hold true for the selection principles $S_{fin}({\tilde{\Phi }}^{\uparrow },{\tilde{\Psi }}^{\uparrow })$.

19 pages