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

TL;DR

This paper explores selection principles related to bornological covers in topological spaces, establishing equivalences, game-theoretic insights, and connections with cardinal invariants, with applications to function spaces and product spaces.

Contribution

It introduces new equivalences and characterizations of selection principles involving bornological covers and their relation to cardinal invariants and tightness properties in function spaces.

Findings

Established equivalences among certain selection principles.

Connected cardinal invariants with base size of bornologies.

Characterized variations of tightness properties via bornological covering properties.

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 $\tau_\mathfrak{B}$ 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 $\mathfrak{B}^n$, $n\in \omega$. 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 $X$.