More on ultrafilters and topological games

Abstract

[EN] Two different open-point games are studied here, the G-game and the Gp-game, defined for each p ∈ ω∗. We prove that for each p ∈ ω∗, there exists a space in which none of the players of the Gp-game has a winning strategy.Nevertheless a result of P. Nyikos, essentially shows that it is consistent, that there exists a countable space in which all these games are undetermined.We construct a countably compact space in which player II of the Gp-game is the winner, for every p ∈ ω∗. With the same technique of construction we built a countably compact space X, such that in X ×X player II of the G-game is the winner. Our last result is to construct ω1-many countably compact spaces, with player I of the G-game as a winner in any countable product of them, but player II is the winner in the product of all of them in the G-game.