Gale-Stewart Theorem

Let be a Game. Suppose that is closed in the Game Space . Then is Determined.

Proof 1

Suppose II has no Winning Game Strategy for . There is a first move such that II has no Winning Game Strategy for starting at . Moreover, if II has no Winning Game Strategy for starting at then for any there is a move such that II has no Winning Game Strategy for starting at . This we obtain a strategy for I. Now suppose they played a game

where I follows the above strategy. Note that if and only if for some and any :

because is closed.

But if this happened, then II would have a Winning Game Strategy post stage , contradicting our construction. We conclude that the above strategy is a I-Winning Game Strategy. Thus is Determined.

Proof 2