On the Use of Infinite Games on Cardinal Function Inequities

For decades, the topic of cardinal functions and the inequalities connecting them have been broadly studied, leading to some notorious results, such as Arhangel'skii's theorem and Hajnal-Juhász' inequality, both providing an upper bound to the cardinality of Hausdorff spaces, as well as Bell, Ginsburg and Woods' theorem, which is a common generalization of the previous two for the class of normal spaces.

Other questions have emerged, such as Arhangel’skii’s problem whether there is an upper bound to Lindelöf spaces with points G_δ and, in recent years, infinite games have been progressively used to give partial answers to problems relating to this field, e.g., Scheepers and Tall’s theorem, which states that if a space has points G_δ and player II has a winning strategy in the Rothberger game of length ω_1, then its cardinality is at most the continuum. In this talk, we want to discuss some of our successes in refining previously known results and inequalities by using the weak Rothberger game and the cellularity game, as well as provide a few examples which illustrate their sharpness whenever possible.

(This is a joint work with Angelo Bella (University of Catania) and Santi Spadaro (University of Catania).)