Borel Determinacy Theorem

Let  be a tree defined on a metric over a set of paths such that the distance between paths  and  is , where  is the number of nodes shared by  and . Let  be a Borel set of paths in the topology induced by this metric. Suppose two players play a game by choosing a path down the tree, so that they alternate and each time choose an immediate successor of the previously chosen point. The first player wins if the chosen path is in . Then one of the players has a winning strategy in this game.