Some applications of the point-open subbase game

Sánchez, D. Guerrero, and Tkachuk, Vladimir Vladimirovich. "Some applications of the point-open subbase game." Commentationes Mathematicae Universitatis Carolinae 58.3 (2017): 383-395. <http://eudml.org/doc/294080>.

@article{Sánchez2017,
abstract = {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 $\lbrace x_n,U_n: n\in ø\rbrace $ have been made and the player $P$ wins if $\bigcup _\{n\in ø\}U_n=X$; 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 $X$.},
author = {Sánchez, D. Guerrero, Tkachuk, Vladimir Vladimirovich},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {point-open game; subbase; winning strategy; players; discrete space; compact space; scattered space; measurable cardinal},
language = {eng},
number = {3},
pages = {383-395},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Some applications of the point-open subbase game},
url = {http://eudml.org/doc/294080},
volume = {58},
year = {2017},
}

TY - JOUR
AU - Sánchez, D. Guerrero
AU - Tkachuk, Vladimir Vladimirovich
TI - Some applications of the point-open subbase game
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2017
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 58
IS - 3
SP - 383
EP - 395
AB - 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 $\lbrace x_n,U_n: n\in ø\rbrace $ have been made and the player $P$ wins if $\bigcup _{n\in ø}U_n=X$; 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 $X$.
LA - eng
KW - point-open game; subbase; winning strategy; players; discrete space; compact space; scattered space; measurable cardinal
UR - http://eudml.org/doc/294080
ER -