## Some applications of the point-open subbase game.(English)Zbl 1424.91025

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 $$\{(x_n,U_n):n\in\omega\}$$ have been made and the player $$P$$ wins if $$\bigcup_{n\in\omega}U_n=X$$; 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 $$X$$.

### MSC:

 91A44 Games involving topology, set theory, or logic 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 91A05 2-person games 54D30 Compactness 54D70 Base properties of topological spaces
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### References:

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