Say that a cardinal number κ is small relative to the space X if κ < ∆(X), where ∆(X) is the least cardinality of a non-empty open set in X . We prove that no Baire metric space can be covered by a small number of discrete sets, and give some generalizations. We show a ZFC example of a regular Baire σ-space and a consistent example of a normal Baire Moore space which can be covered by a small n...

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 $G(X)$....

When studying questions about real numbers, it is common practice in set theory to investigate the closely related Baire space ω and Cantor space 2 . These spaces have been extensively studied by set theorists from various points of view, e.g., questions about cardinal characteristics of the continuum, descriptive set theory and other combinatorial questions. Furthermore, the investigation of 2...