Banach game

In mathematics, the Banach game is a topological game introduced by Stefan Banach in 1935 in the second addendum to problem 43 of the Scottish book as a variation of the Banach–Mazur game.^[1]

Given a subset $X$ of real numbers, two players alternatively write down arbitrary (not necessarily in $X$ ) positive real numbers $x_{0},x_{1},x_{2},\ldots$ such that $x_{0}>x_{1}>x_{2}>\cdots$ Player one wins if and only if $\sum _{i=0}^{\infty }x_{i}$ exists and is in $X$ .^[2]

One observation about the game is that if $X$ is a countable set, then either of the players can cause the final sum to avoid the set.^[3] Thus in this situation the second player has a winning strategy.

References[edit]

^ Mauldin, R. Daniel (April 1981). The Scottish Book: Mathematics from the Scottish Cafe (1 ed.). Birkhäuser. p. 113. ISBN 978-3-7643-3045-3.
^ Telgársky, Rastislav (Spring 1987). "Topological Games: On the 50th Anniversary of the Banach–Mazur Game" (PDF). Rocky Mountain Journal of Mathematics. 17 (2): 227–276. at 242.
^ Mauldin 1981, p. 116.

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