Original scientific paper

https://doi.org/10.3336/gm.43.1.14

One more variation of the point-open game

Arcos Daniel Jardón ; Departamento de Matemáticas, Universidad Autónoma Metropolitana
Vladimir V. Tkachuk ; Departamento de Matemáticas, Universidad Autónoma Metropolitana

Abstract

A topological game "Dense Gδσ-sets" (also denoted by DG) is introduced as follows: for any n ω at the n-th move the player I takes a point xn v X and II responds by taking a Gδ-set Qn in the space X such that xn Qn. The play stops after ω moves and I wins if the set {Qn : 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 Cp(X); one of the applications is that, for a Lindelöf Σ-space X, the space Cp(X) is I-favorable for DG if and only if X is ω-monolithic.

Keywords

Topological game; player; winning strategy; dense Gδσ-sets; separable space; dyadic compact space; scattered compact space; neutral space; function space

Hrčak ID:

23541

URI

https://hrcak.srce.hr/23541

Publication date:

25.5.2008.

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