Classes of Baire spaces defined by topological games

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