Summer Conference on Topology and Its Applications

Document Type

Topology + Foundations

Publication Date


Publication Source

32nd Summer Conference on Topology and Its Applications


Given a subbase S of a space X, the game PO(S,X) is defined for two players P and O who respectively pick, at the n-th move, a point xn 2 X and a set Un 2 S such that xn 2 Un . The game stops after the moves {xn, Un : n 2 !} have been made and the player P wins if the union of the Un’s equals X; otherwise O is the winner. Since PO(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(S,X) for a given subbase S. It turns out that, for any subbase S, the player P has a winning strategy in PO(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 S such that neither P nor O has a winning strategy in the game PO(S,X). Given a compact space X, we show that the games PO(S,X) and PO(X) are equivalent for any subbase S of the space X.


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