Transfinite Games and Regularity Properties in Higher Baire Spaces
Orestis Tsakakos

Abstract:
Since their formal introduction in 1953 by Gale and Stewart [GS53], infinite games of length ω have been a central part of the study of the continuum. However, games of uncountable length have received limited attention partly because of the inconsistency of the axiom of determinacy (AD) with ZF, and the fact that many determinacy results fail in higher cardinalities, even for very simple sets. However, with the rise of generalized descriptive set theory, specific types of transfinite games corresponding to regularity properties have been studied in the context of higher Baire spaces, such as the Banach-Mazur game and the perfect set game.
This thesis is a rigorous analysis of transfinite games in higher Baire spaces κ^κ for an uncountable cardinal κ satisfying κ^{<κ} = κ. The underlying goal is to provide determinacy results for simple subsets of the spaces κ^κ and 2^κ in ZFC, while also trying to isolate specific properties of games that are connected to their determinacy.
First, we review some essential facts about standard games in the transfinite setting. While the Gale-Stewart theorem (i.e., closed and open determinacy of standard games) fails in higher Baire spaces, we show that an analogue of this does hold if the set of non-losing positions is sufficiently closed either for Player I or Player II.
Second, we discuss the class of asymmetric games first introduced by Kechris in [Kec77] and then generalized to the transfinite setting in [SSz], in which Player I plays elements of κ^{<κ} , while II imposes requirements that Player I must satisfy in the next round. The determinacy of such games for sets definable from a κ-sequence of ordinals is consistent, relative to an inaccessible cardinal, by results of [SSz]. We show that Kechris’s games can be coded as instances of the standard game via a function which is continuous in a strong sense.
The majority of our new results concern variants of Kechris’s games, where Player I can now only play sequences whose length is bounded by some µ < κ. We prove the existence of closed non-determined subsets of κ^κ for games of this form. Our results confirm the intuition that determinacy of such games is closely related to whether or not Player I can play sequences of any length less than κ. Furthermore, they provide deeper insight on how games behave in the transfinite setting.