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 $$X$$ 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 $$\sum^\infty_{i=0} x_i$$ 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 has a winning strategy.