Some applications of the point-open subbase game

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 $X$