Banach game

Template:Short description 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.

Given a subset $X$ of real numbers, two players alternatively write down arbitrary (not necessarily in $X$) positive real numbers $x_0,x_1,x_2,\dots$ such that $x_0>x_1>x_2>\cdots$ Player one wins if and only if $\sum _{i=0}^{\mathrm{\infty }}{x}_i$ exists and is in $X$.

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.Template:Sfn Thus in this situation the second player has a winning strategy.

Further reading

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