Topological games and continuity of group operations

We consider a topological game GΠ involving two players α and β and show that, for a paratopological group, the absence of a winning strategy for player β implies the group is a topological one. We provide a large class of topological spaces X for which the absence of a winning strategy for player β is equivalent to the requirement that X is a Baire space. This allows to extend the class of paratopological or semitopological groups for which one can prove that they are, actually, topological groups.Conditions of the type “existence of a winning strategy for the player α” or “absence of a winning strategy for the player β” are frequently used in mathematics. Though convenient and satisfactory for theoretical considerations, such conditions do not reveal much about the internal structure of the topological space where they hold. We show that the existence of a winning strategy for any of the players in all games of Banach-Mazur type can be expressed in terms of “saturated sieves” of open sets.