Certain Observations on Tightness and Topological Games in Bornology

Abstract

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. in Indag Math 31:43-63, 2020; Das et al. in Topol Appl 310:108005, 2022) using the idea of strong uniform convergence (Beer and Levi in J Math Anal Appl 350:568-589, 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 the supertightness, the Id-fan tightness and the T-tightness. Certain situations are discussed when C(X) is a 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, the Gruenhage game on \((C(X),\tau ^s_\mathfrak {B})\) and certain games on X are presented.