Topological games and productively countably tight spaces

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.