Every infinite game of perfect information on a set-sized move alphabet with a Borel payoff in its discrete product topology has a winning strategy in an infinite game for one player. A proof strengthens the induction hypothesis to existence of arbitrary-depth-preserving game coverings unravelling the payoff. The closed-set announcement construction starts the induction. For countable unions, successive coverings stabilize each finite level and their inverse limit makes every constituent payoff clopen; their union is open and can be unravelled once more.
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The Borel determinacy theorem is a significant result in set theory, particularly in the context of descriptive set theory. It concerns games played on sets of natural numbers and specifically establishes that certain types of games are determined.