Some new versions of an old game

Vladimir Vladimirovich Tkachuk

Commentationes Mathematicae Universitatis Carolinae (1995)

  • Volume: 36, Issue: 1, page 177-196
  • ISSN: 0010-2628

Abstract

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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 ∈ X and the second responds with choosing an open U n ∋ x n . The game stops after ω moves and the first player wins if ∪ { U n : n ∈ ω } = X . Otherwise the victory is ascribed to the second player. In this paper we introduce and study the games θ and Ω . In θ the moves are made exactly as in the point-open game, but the first player wins iff ∪ { U n : n ∈ ω } is dense in X . In the game Ω the first player also takes a point x n ∈ X at his (or her) n -th move while the second picks an open U n ⊂ X with x n ∈ U ¯ n . The conclusion is the same as in θ , i.eṫhe first player wins iff ∪ { U n : n ∈ ω } is dense in X . It is clear that if the first player has a winning strategy on a space X for the game θ or Ω , then X is in some way similar to a separable space. We study here such spaces X calling them θ -separable and Ω -separable respectively. Examples are given of compact spaces on which neither θ nor Ω are determined. It is established that first countable θ -separable (or Ω -separable) spaces are separable. We also prove that 1) all dyadic spaces are θ -separable; 2) all Dugundji spaces as well as all products of separable spaces are Ω -separable; 3) Ω -separability implies the Souslin property while θ -separability does not.

How to cite

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Tkachuk, Vladimir Vladimirovich. "Some new versions of an old game." Commentationes Mathematicae Universitatis Carolinae 36.1 (1995): 177-196. <http://eudml.org/doc/247707>.

@article{Tkachuk1995,
abstract = {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 $\cup \lbrace U_n:n\in \omega \rbrace =X$. 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 \lbrace U_n:n\in \omega \rbrace $ 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.eṫhe first player wins iff $\cup \lbrace U_n:n\in \omega \rbrace $ 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 1) all dyadic spaces are $\theta $-separable; 2) all Dugundji spaces as well as all products of separable spaces are $\Omega $-separable; 3) $\Omega $-separability implies the Souslin property while $\theta $-separability does not.},
author = {Tkachuk, Vladimir Vladimirovich},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {topological game; strategy; separability; $\theta $-separability; $\Omega $-separability; point-open game; Galvin-Telgársky point-open game},
language = {eng},
number = {1},
pages = {177-196},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Some new versions of an old game},
url = {http://eudml.org/doc/247707},
volume = {36},
year = {1995},
}

TY - JOUR
AU - Tkachuk, Vladimir Vladimirovich
TI - Some new versions of an old game
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1995
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 36
IS - 1
SP - 177
EP - 196
AB - 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 $\cup \lbrace U_n:n\in \omega \rbrace =X$. 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 \lbrace U_n:n\in \omega \rbrace $ 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.eṫhe first player wins iff $\cup \lbrace U_n:n\in \omega \rbrace $ 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 1) all dyadic spaces are $\theta $-separable; 2) all Dugundji spaces as well as all products of separable spaces are $\Omega $-separable; 3) $\Omega $-separability implies the Souslin property while $\theta $-separability does not.
LA - eng
KW - topological game; strategy; separability; $\theta $-separability; $\Omega $-separability; point-open game; Galvin-Telgársky point-open game
UR - http://eudml.org/doc/247707
ER -

References

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