Certain observations on tightness and topological games in bornology

Debraj Chandra, Pratulananda Das, Subhankar Das

Published 2022-09-03Version 1

This article is a continuation of our investigations in the function space $C(X)$ with respect to the topology $\tau^s_{\mathfrak{B}}$ of strong uniform convergence on $\mathfrak{B}$ in line of (Chandra et al. 2020 and Das et al. 2022 ) using the idea of strong uniform convergence (Beer and Levi, 2009 ) on a bornology. First we focus on the notion of the tightness property of $(C(X),\tau^s_{\mathfrak{B}})$ and some of its variations such as supertightness, Id-fan tightness and $T$-tightness. Certain situations are discussed when $C(X)$ is a {\rm k}-space with respect to the topology $\tau^s_{\mathfrak{B}}$. Next the notions of strong $\mathfrak{B}$-open game and $\gamma_{{\mathfrak{B}}^s}$-open game on $X$ are introduced and some of its consequences are investigated. Finally, we consider discretely selective property and related games. On $(C(X),\tau^s_{\mathfrak{B}})$ several interactions between topological games related to discretely selective property including the Gruenhage game on $(C(X),\tau^s_{\mathfrak{B}})$ are presented.