Spaces of Real Functions, Covers and Dense Subsets

Abstract: In paper we study relationships between covering properties of a topological space $X$ and the space $(USC^*(X),\tau_{\mathcal{B}})$ of bounded upper semicontinuous functions on $X$ with the topology $\tau_{\mathcal{B}}$ defined by the bornology $\mathcal{B}$ on $X$. We present characterizations of various local properties of $(USC^*(X),\tau_{\mathcal{B}})$ in terms of selection principles related to bornological covers of $X$. We also investigate topological properties of sequences of upper dense and upper sequentially dense subsets of $(USC^*(X),\tau_{\mathcal{B}})$.