Certain results on selection principles associated with bornological structure in topological spaces

Abstract

We study selection principles related to bornological covers in a topological space $X$ following the work of Aurichi et al., 2019, where selection principles have been investigated in the function space $C_{\mathfrak{B}}(X)$ endowed with the topology $\ddot{I}{„}_{\mathfrak{B}}$ of uniform convergence on bornology $\mathfrak{B}$. We show equivalences among certain selection principles and present some game theoretic observations involving bornological covers. We investigate selection principles on the product space $X^n$ equipped with the product bornolgy ${\mathfrak{B}}^n$, $n\in \ddot{I}‰$. Considering the cardinal invariants such as the unbounding number ($\mathfrak{b}$), dominating numbers ($\mathfrak{d}$), pseudointersection numbers ($\mathfrak{p}$) etc., we establish connections between the cardinality of base of a bornology with certain selection principles. Finally, we investigate some variations of the tightness properties of $C_{\mathfrak{B}}(X)$ and present their characterizations in terms of selective bornological covering properties of $X$.