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 , … such that x 0 > x 1 > x 2 > ⋯ Player one wins if and only if ∑ i = 0 ∞ 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. Thus in this situation the second player can win.
