Further observations on bornological covering properties and selection principles

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 $(C(X),τ^s_\mathfrak{B})$ like countable fan tightness, countable strong fan tightness along with the Reznichenko's property.

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