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On point-picking games. (English) Zbl 0604.54006
Let X be a topological space, P a property of subsets of X and $$\alpha$$ an ordinal. We consider a topological two person game $$G^ p_{\alpha}(X)$$ as follows. Player I chooses an open set U of X, and player II chooses a point x such that $$x\in U$$. This play is repeated less than $$\alpha$$-times. Player I wins if the set of points picked by player II has property P, otherwise player II wins. The main result reads as follows: For any cardinal $$\kappa \leq 2^{\omega}$$, the following are equivalent: (1) There exists a $$T_ 3$$-space X with $$\delta (X)=\omega$$ and $$\pi (X)=\kappa$$ for which player II wins $$G^ D_{\omega}(X)$$, where D denotes the property being dense; (2) the real line can be written as the union of $$\leq \kappa$$ nowhere dense subsets. Under the assumption: $$\delta (x)=\omega$$ and $$\pi (x)=\kappa$$ on X and if the real line is not the union of $$\leq \kappa$$ nowhere dense sets, namely Martin’s axiom for countable posets and $$\leq \kappa$$ dense sets holds, then player I wins this game.
Reviewer: K.Iseki

##### MSC:
 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 91A24 Positional games (pursuit and evasion, etc.) 54F65 Topological characterizations of particular spaces
##### Keywords:
topological game; property of subsets; real line