| Title:
|
Some new versions of an old game (English) |
| Author:
|
Tkachuk, Vladimir V. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
36 |
| Issue:
|
1 |
| Year:
|
1995 |
| Pages:
|
177-196 |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
topological game |
| Keyword:
|
strategy |
| Keyword:
|
separability |
| Keyword:
|
$\theta$-separability |
| Keyword:
|
$\Omega$-separability |
| Keyword:
|
point-open game |
| MSC:
|
03E50 |
| MSC:
|
54A35 |
| MSC:
|
90D40 |
| idZBL:
|
Zbl 0838.54005 |
| idMR:
|
MR1334425 |
| . |
| Date available:
|
2009-01-08T18:16:53Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118743 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
|
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| Reference:
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| . |
