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On a game of Sierpiński. (English) Zbl 0671.90110

Let Y be a topological space, \(X\subseteq Y\). The Sierpiński game S(X,Y) proceeds as follows. Player I chooses an \(A_ 1\subseteq X\). If it is uncountable, player II answers by an uncountable \(B_ 1\subseteq A_ 1\), otherwise one sets \(B_ 1=\emptyset\). Then player I chooses an \(A_ 2\subseteq B_ 1\) and the moves continue under the same condition. Player II is declared winner if \(\cap \bar B_ n\subseteq X\). The author presents several interesting theorems concerning the determinacy and winning strategies under special conditions on X and Y. Also, a related Banach-Mazur game is discussed.

MSC:

91A99 Game theory
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
91A05 2-person games
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