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

One observation about the game is that if ${\displaystyle X}$ 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.