Glasnik Matematicki, Vol. 43, No.1 (2008), 205-217.

ONE MORE VARIATION OF THE POINT-OPEN GAME

Arcos Daniel Jardón and Vladimir V. Tkachuk

Abstract.   A topological game "Dense G_δσ-sets" (also denoted by DG) is introduced as follows: for any n ω at the n-th move the player I takes a point x_n v X and II responds by taking a G_δ-set Q_n in the space X such that x_n Q_n. The play stops after ω moves and I wins if the set {Q_n : n ω} is dense in X. Otherwise the player II is declared to be the winner. We study classes of spaces on which the player I has a winning strategy. It is evident that the I-favorable spaces constitute a generalization of the class of separable spaces. We show that there exists a neutral space for the game DG and prove, among other things, that Lindelöf scattered spaces and dyadic spaces are I-favorable. We characterize I-favorability for the game DG in the spaces C_p(X); one of the applications is that, for a Lindelöf Σ-space X, the space C_p(X) is I-favorable for DG if and only if X is ω-monolithic.

2000 Mathematics Subject Classification.   54H11, 54C10, 22A05, 54D25, 54C25.

Key words and phrases.   Topological game, player, winning strategy, dense G_δσ-sets, separable space, dyadic compact space, scattered compact space, neutral space, function space.

DOI: 10.3336/gm.43.1.14

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