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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

54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
91A24 Positional games (pursuit and evasion, etc.)
54F65 Topological characterizations of particular spaces