Borel Determinacy Theorem

Let $T$ be a tree defined on a metric over a set of paths such that the distance between paths $p$ and $q$ is $1/n$, where $n$ is the number of nodes shared by $p$ and $q$. Let $A$ 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 $A$. Then one of the players has a winning Strategy in this Game.

See also Game Theory, Strategy