One more variation of the point-open game

Abstract

A topological game Dense G��-sets (also denoted by DG) is introduced as follows: for any n 2 ! at the n-th move the player I takes a point xn 2 X and II responds by taking a G�-set Qn in the space X such that xn 2 Qn. The play stops after ! moves and I wins if the set S {Qn : n 2 !} 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 Lindelof scattered spaces and dyadic spaces are I-favorable. We characterize I- favorability for the game DG in the spaces Cp(X); one of the applications is that, for a Lindelof �-space X, the space Cp(X) is I-favorable for DG if and only if X is !-monolithic.