Classes of Baire spaces defined by topological games

TL;DR

This paper introduces new classes of Baire spaces defined via topological games, explores their properties, and applies these concepts to the study of continuity in topological groups.

Contribution

It defines and analyzes classes of Baire spaces based on modified topological games, including a four-player variant, linking game-theoretic properties to topological group operations.

Findings

Equivalent games are identified to simplify analysis.

Classes of spaces are characterized by game-theoretic properties.

Applications to continuity of operations in topological groups.

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.