Topological games and productively countably tight spaces

TL;DR

This paper explores the relationship between topological game strategies and the property of productive countable tightness in spaces, establishing conditions under which these properties are equivalent or imply each other.

Contribution

It introduces new characterizations linking game-theoretic strategies to productive countable tightness, extending previous results by Uspenskii and Scheepers.

Findings

Player II's winning strategy implies productive countable tightness

Productively countably tight spaces satisfy a specific selection principle

Several corollaries follow from the main characterizations

Abstract

The two main results of this work are the following: if a space $X$ is such that player II has a winning strategy in the game $\gone(\Omega_x, \Omega_x)$ for every $x \in X$, then $X$ is productively countably tight. On the other hand, if a space is productively countably tight, then $\sone(\Omega_x, \Omega_x)$ holds for every $x \in X$. With these results, several other results follow, using some characterizations made by Uspenskii and Scheepers.