Vladimir V. Tkachuk
Some new versions of an old game

Comment.Math.Univ.Carolinae 36,1 (1995) 179-198.

Abstract:The old game is the point-open one discovered independently by F. Galvin [7] and R. Telg\'arsky [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. \par 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 $X$. \par 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.

Keywords: topological game, strategy, separability, $\theta $-separability, $\Omega $-separability, point-open game
AMS Subject Classification: 03E50, 54A35

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