Classes of Baire spaces defined by topological games

Abstract

The article studies topological games that arise in the study of the continuity of operations in groups with topology, such as paratopological and semitopological groups. These games are modifications of the Banach--Mazur game. Given a two-player game $G(X)$ of the Banach--Mazur type, we define $\Gamma^G$-Baire, $\Gamma^G$-nonmeager and $\Gamma^G$-spaces. A space $X$ is a $\Gamma^G$-Baire if the second player does not have a winning strategy in $G(X)$. The classes of $\Gamma^G$-nonmeager spaces and $\Gamma^G$-spaces are defined similarly, with the help of modifications of the game $G(X)$. For the games under consideration, equivalent games are found, which facilitates studying the relationship between the resulting classes of spaces and determining which spaces belong to these classes. For this purpose, we introduce a modification of the Banach--Mazur game with four players. Results of this paper find application in the study the continuity of operations in groups with topology.