Open determinacy

An infinite game of perfect information with an open set of winning plays for I in the product topology on a discrete move set has a winning strategy in an infinite game for one player. Construct a winning-position attractor in an infinite game by transfinite recursion; ranks give I a terminating descent strategy inside it, and its complement gives II a strategy avoiding every winning prefix. The move set need not be countable, in the usual set theory with axiom of choice.

Start with positions whose entire extension cylinder set lies in the winning open set. At each successor stage add I-positions with some successor already present and II-positions with every successor already present; take unions at limit ordinals. The stable set is the least fixed point of this order-preserving operation. The first entry ordinal supplies a strictly decreasing rank until the winning prefix is reached.