Some new versions of an old game

Abstract

summary:The old game is the point-open one discovered independently by F. Galvin [7] and R. Telgársky [17]. Recall that it is played on a topological space $X$ as follows: at the $n$-th move the first player picks a point $x_n\in X$ and the second responds with choosing an open $U_n\ni x_n$. The game stops after $\omega$ moves and the first player wins if $\cup\{U_n:n\in\omega\}=X$. Otherwise the victory is ascribed to the second player. In this paper we introduce and study the games $\theta$ and $\Omega$. In $\theta$ the moves are made exactly as in the point-open game, but the first player wins iff $\cup\{U_n:n\in\omega\}$ is dense in $X$. In the game $\Omega$ the first player also takes a point $x_n\in X$ at his (or her) $n$-th move while the second picks an open $U_n\subset X$ with $x_n\in\overline{U}_n$. The conclusion is the same as in $\theta$, i.e\. the first player wins iff $\cup\{U_n:n\in\omega\}$ is dense in $X$. It is clear that if the first player has a winning strategy on a space $X$ for the game $\theta$ or $\Omega$, then $X$ is in some way similar to a separable space. We study here such spaces $X$ calling them $\theta$-separable and $\Omega$-separable respectively. Examples are given of compact spaces on which neither $\theta$ nor $\Omega$ are determined. It is established that first countable $\theta$-separable (or $\Omega$-separable) spaces are separable. We also prove that \newline 1) all dyadic spaces are $\theta$-separable; \newline 2) all Dugundji spaces as well as all products of separable spaces are $\Omega$-separable; \newline 3) $\Omega$-separability implies the Souslin property while $\theta$-separability does not