Spaces of Real Functions, Covers and Dense Subsets

TL;DR

This paper explores how covering properties of a topological space relate to the structure and properties of the space of bounded upper semicontinuous functions, using bornology-based topologies and selection principles.

Contribution

It provides new characterizations of local properties of the function space in terms of bornological covers and selection principles, linking topological and functional properties.

Findings

Characterizations of local properties via bornological covers

Topological properties of dense subsets in function spaces

Relationships between covering properties and function space topology

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 $(USC^*(X),\tau_{\mathcal{B}})$.