| Title:
|
Some applications of the point-open subbase game (English) |
| Author:
|
Sánchez, D. Guerrero |
| Author:
|
Tkachuk, V. V. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
58 |
| Issue:
|
3 |
| Year:
|
2017 |
| Pages:
|
383-395 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Given a subbase $\mathcal S$ of a space $X$, the game $PO(\mathcal S,X)$ is defined for two players $P$ and $O$ who respectively pick, at the $n$-th move, a point $x_n\in X$ and a set $U_n\in \mathcal S$ such that $x_n\in U_n$. The game stops after the moves $\{x_n,U_n: n\in\o\}$ have been made and the player $P$ wins if $\bigcup_{n\in\o}U_n=X$; otherwise $O$ is the winner. Since $PO(\mathcal S,X)$ is an evident modification of the well-known point-open game $PO(X)$, the primary line of research is to describe the relationship between $PO(X)$ and $PO(\mathcal S,X)$ for a given subbase $\mathcal S$. It turns out that, for any subbase $\mathcal S$, the player $P$ has a winning strategy in $PO(\mathcal S,X)$ if and only if he has one in $PO(X)$. However, these games are not equivalent for the player $O$: there exists even a discrete space $X$ with a subbase $\mathcal S$ such that neither $P$ nor $O$ has a winning strategy in the game $PO(\mathcal S,X)$. Given a compact space $X$, we show that the games $PO(\mathcal S,X)$ and $PO(X)$ are equivalent for any subbase $\mathcal S$ of the space $X$. (English) |
| Keyword:
|
point-open game |
| Keyword:
|
subbase |
| Keyword:
|
winning strategy |
| Keyword:
|
players |
| Keyword:
|
discrete space |
| Keyword:
|
compact space |
| Keyword:
|
scattered space |
| Keyword:
|
measurable cardinal |
| MSC:
|
54A25 |
| MSC:
|
54D30 |
| MSC:
|
54D70 |
| MSC:
|
91A05 |
| idZBL:
|
Zbl 06837073 |
| idMR:
|
MR3708781 |
| DOI:
|
10.14712/1213-7243.2015.210 |
| . |
| Date available:
|
2017-11-22T09:26:39Z |
| Last updated:
|
2019-10-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146904 |
| . |
| Reference:
|
[1] Baldwin S.: Possible point-open types of subsets of the reals.Topology Appl. 38 (1991), 219–223. Zbl 0719.54003, MR 1098902, 10.1016/0166-8641(91)90087-3 |
| Reference:
|
[2] Daniels P., Gruenhage G.: The point-open types of subsets of the reals.Topology Appl. 37 (1990), no. 1, 53–64. MR 1075373, 10.1016/0166-8641(90)90014-S |
| Reference:
|
[3] Engelking R.: General Topology.PWN, Warszawa, 1977. Zbl 0684.54001, MR 0500780 |
| Reference:
|
[4] Galvin F.: Indeterminacy of point-open games.Bull. Acad. Polon. Sci. Sér. Math. 26 (1978), no. 5, 445–449. Zbl 0392.90101, MR 0493925 |
| Reference:
|
[5] Kuratowski K.: Topology, Volume 1.Academic Press, New York, 1966. MR 0217751 |
| Reference:
|
[6] Laver R.: On the consistency of Borel's conjecture.Acta Math. 137 (1976), 151–169. Zbl 0357.28003, MR 0422027, 10.1007/BF02392416 |
| Reference:
|
[7] Pawlikowski J.: Undetermined sets of point-open games.Fund. Math. 144 (1994), 279–285. Zbl 0853.54033, MR 1279482 |
| Reference:
|
[8] Telgársky R.: Spaces defined by topological games.Fund. Math. 88 (1975), 193–223. MR 0380708, 10.4064/fm-88-3-193-223 |
| Reference:
|
[9] Telgársky R.: Spaces defined by topological games, II.Fund. Math. 116 (1983), no. 3, 189–207. Zbl 0558.54029, MR 0716219, 10.4064/fm-116-3-189-207 |
| . |
