Applications of Bornological Covering Properties in Metric Spaces

Abstract

Using the idea of strong uniform convergence on bornology, Caserta, Di Maio and Ko\v{c}inac studied open covers and selection principles in the realm of metric spaces (associated with a bornology) and function spaces (w.r.t. the topology of strong uniform convergence). We primarily continue in the line initiated before and investigate the behaviour of various selection principles related to these classes of bornological covers. In the process we obtain implications among these selection principles resulting in Scheepers' like diagrams. We also introduce the notion of strong-$\mathfrac{B}$-Hurewicz property and investigate some of its consequences. Finally, in $C(X)$ with respect to the topology $\tau_{\mathfrac{B}}^s$ of strong uniform convergence, important properties like countable $T$-tightness, Reznichenko property are characterized in terms of bornological covering properties of $X$.