Further observations on bornological covering properties and selection principles

TL;DR

This paper extends the study of bornological covering properties and selection principles in metric spaces, providing new characterizations, properties, and relationships involving bornology, Ramsey relations, and product spaces.

Contribution

It introduces new characterizations of selection principles related to bornological covers and explores properties of bornological spaces and their finite powers.

Findings

Characterizations of selection principles via Ramseyan partition relations.

New observations on the $rak{B}^s$-Hurewicz and $rak{B}^s$-Gerlits-Nagy properties.

Characterizations of properties of $(C(X), au^s_rak{B})$ such as tightness and Reznichenko's property.

Abstract

This article is a continuation of the study of bornological open covers and related selection principles in metric spaces done in (Chandra et al. 2020) using the idea of strong uniform convergence (Beer and Levi, 2009) on bornology. Here we explore further ramifications, presenting characterizations of various selection principles related to certain classes of bornological covers using the Ramseyan partition relations, interactive results between the cardinalities of bornological bases and certain selection principles involving bornological covers, producing new observations on the $\mathfrak{B}^s$-Hurewicz property introduced in (Chandra et al. 2020) and several results on the $\mathfrak{B}^s$-Gerlits-Nagy property of $X$ which is introduced here following the seminal work of (Gerlits and Nagy, 1982). In addition, in the finite power $X^n$ with the product bornology $\mathfrak{B}^n$, the ${\mathfrak{B}^n}^s$-Hurewicz property as well as the ${\mathfrak{B}^n}^s$-Gerlits-Nagy property of $X^n$ are characterized in terms of properties of $(C(X),\tau^s_\mathfrak{B})$ like countable fan tightness, countable strong fan tightness along with the Reznichenko's property.