Gale and Stewart (1953) proved that if the payoff set is an open or closed subset of A&omega; then the Gale-Stewart game with that payoff set is always determined.
Harvey Friedman (1971) proved that any proof that all Borel subsets of Cantor space ({0,1}ω ) were determined would require repeated use of the axiom of replacement, an axiom not typically required to prove theorems about "small" objects such as Cantor space.