Certain observations on tightness and topological games in bornology

TL;DR

This paper investigates the topological properties and games related to the function space $C(X)$ with strong uniform convergence topology on a bornology, extending previous work and exploring new game-theoretic and tightness concepts.

Contribution

It introduces new notions of tightness, topological games, and their interactions in $C(X)$ with the strong uniform convergence topology on a bornology, extending prior research.

Findings

Characterized tightness and supertightness in $C(X)$

Analyzed conditions for $C(X)$ to be a k-space under the topology

Explored relationships between topological games and discretely selective properties

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. 2020 \cite{dcpdsd} and Das et al. 2022 \cite{pddcsd-3}) using the idea of strong uniform convergence (Beer and Levi, 2009 \cite{bl}) on a bornology. First we focus on the notion of 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, the Gruenhage game on $(C(X),\tau^s_\mathfrak{B})$ and certain games on $X$ are presented.