Some applications of the point-open subbase game

[摘要] 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~$X$.

[学科分类] 物理化学和理论化学

[关键词] point-open game;subbase;winning strategy;players;discrete space;compact space;scattered space;measurable cardinalDOI: DOI 10.14712/1213-7243.2015.210AMS Subject Classification: 54A25 91A05 54D30 54D70 PDF