Certain results on selection principles associated with bornological structure in topological spaces
Debraj Chandra, Subhankar Das, Nur Alam
TL;DR
The paper develops a systematic framework connecting bornologies, selective covering principles, and function-space topology. It establishes a web of equivalences among selection principles ($\mathsf{S}_1$, $\mathsf{S}_{\mathrm{fin}}$, $\mathsf{U}_{\mathrm{fin}}$) for bornological covers, analyzes their stability under product spaces and continuous images, and links these to cardinal invariants such as $\mathfrak{b}$, $\mathfrak{d}$, $\mathfrak{p}$, and $\mathrm{cov}(\mathcal{M})$. The work further characterizes how bornological covering properties of $X$ determine local and global properties of the function space $C_{\mathfrak{B}}(X)$ under the topology $\tau_{\mathfrak{B}}$, including Reznichenko, countable strong fan tightness, and Fréchet-Urysohn-type properties, with tightness and supertightness relations extended to product spaces. Overall, the paper broadens the interaction between bornology, selection principles, and the topology of function spaces, offering tools for analyzing covering properties and tightness phenomena in both $X$ and $C_{\mathfrak{B}}(X)$.\n
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 $τ_\mathfrak{B}$ 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 $\mathfrak{B}^n$, $n\in ω$. 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 $X$.