Banach game
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.
Given a subset <math>X</math> of real numbers, two players alternately write down arbitrary (not necessarily in <math>X</math>) positive real numbers <math>x_0, x_1, x_2,\ldots</math> such that <math>x_0 > x_1 > x_2 >\cdots</math> Player one wins if and only if <math>\sum^\infty_{i=0} x_i</math> exists and is in <math>X</math>.
One observation about the game is that if <math>X</math> is a countable set, then either of the players can cause the final sum to avoid the set. Thus in this situation the second player has a winning strategy.