 Unravelling of a game payoff

A covering of an infinite game unravels a payoff when its inverse image is a clopen subset of the new infinite-play space. A closed payoff can be unravelled by letting Player I announce a subset of the first exit positions, and letting Player II accept that announcement or challenge one announced position. Accepted plays stop at exit positions, while challenged plays are forced through the challenged exit. Strategy-transfer maps make this more than an ordinary continuous representation of the payoff. Depth-stabilizing inverse limits of coverings handle countable unions in a transfinite induction.