# Applications of Bornological Covering Properties in Metric Spaces

## TL;DR

This paper explores how bornological covering properties influence selection principles and topological features in metric and function spaces, extending previous work with new properties and implications.

## Contribution

It introduces the strong-$ ext{B}$-Hurewicz property and characterizes key topological properties of function spaces via bornological covering properties.

## Key findings

- Implications among selection principles form Scheepers-like diagrams.
- Introduction of strong-$ ext{B}$-Hurewicz property and its consequences.
- Characterization of countable $T$-tightness and Reznichenko property in $C(X)$.

## 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 $X$.