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Some new versions of an old game. (English) Zbl 0838.54005
The author introduces and investigates two variants of Galvin-Telgársky point-open game.
The game \(\Theta\) is played by player I choosing point \(x_n\) and by player II choosing an open set \(U_n \ni x_n\) of a topological space \(X\). The point-player wins if \(\bigcup U_n\) is dense in \(X\). The space \(X\) is said to be \(\Theta\)-separable if the point-player has a winning strategy \((\Theta\)-anti-separable if his/her opponent does). The game \(\Omega\) is defined similarly, the moves satisfying weaker condition \(x_n \in \overline U_n\).
Many different results on \(\Theta (\Omega)\)-(anti)separable spaces are obtained corresponding to dyadic, Dugundji, Souslin, weakly Lindelöf, first-countable, Eberlein compact spaces and many others. The paper ends with a list of 30 open problems.

54A35 Consistency and independence results in general topology
03E50 Continuum hypothesis and Martin’s axiom
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