Compact families and continuity of the inverse (Q2815290)

\((G,\cdot,\tau)\) is a paratopological group if \((G,\cdot)\) is a group, \((G,\tau)\) is a topological space and the binary operation is continuous. \textit{P. Kenderov, I. Kortezov} and \textit{W. Moors} in [Topology Appl. Vol.109, No. 2, 157--165 (2001; Zbl 0976.22003)] introduced a game \(\mathcal G_S(D)\) such that the nonexistence of a winning strategy for one of the players is a sufficient condition for a paratopological group to be topological (i.e., that the inverse operation is also continuous). The author modifies the winning condition of the game \(\mathcal G_S(D)\), obtaining two new games, \(\mathcal G(X)\) and \(\mathcal G(D)\). For each of them, he proves again that the nonexistence of a winning strategy for one of the players implies that a given paratopological group is topological.