Topological games and continuity of group operations (Q712206)

It is known that a Hausdorff paratopological group or semitopological group satisfying a topological property, for example a natural compactness type property, turns out to be a topological group. In this paper, the authors introduce a topological game \(G_{\varPi}\) (played by players \(\alpha\) and \(\beta\)) and show that if in a paratopological group the player \(\beta\) does not have a winning strategy in the game \(G_{\varPi}\), then the group is a topological group. Furthermore, they define a ``relaxed game \(G^{\sim}\)'' for a Baire space \(X\) and a game \(G\) and prove that if \(\alpha\) has a winning strategy in \(G^{\sim}\), then \(\beta\) does not have a winning strategy in \(G\). They show that this result allows to describe a very large class of topological spaces for which every Hausdorff paratopological or semitopological group turns out to be a topological group. In section 5, they show that the existence of a winning strategy for player \(\alpha\) in a Banach-Mazur-type game \(G\) is equivalent to the presence in the space of a ``saturated sieve'' of open sets with decreasing ``sieve sequences'' having some additional property determined by the winning rule of the game \(G\) and the player \(\beta\) has a winning strategy in the game \(G\) if and only if some open subset of the space admits a saturated sieve where none of the sieve sequences has the property related to the winning rule of \(G\).