G_δ-sets in topological spaces and games

Players ONE and TWO play the following game: In the nth inning ONE chooses a set O_n from a prescribed family ℱ of subsets of a space X; TWO responds by choosing an open subset T_n of X. The players must obey the rule that O_n ⊆ O_n+1 ⊆ T_n+1 ⊆ T_n for each n. TWO wins if the intersection of TWO's sets is equal to the union of ONE's sets. If ONE has no winning strategy, then each element of ℱ is a G_δ-set. To what extent is the converse true? We show that: (A) For ℱ the collection of countable subsets of X: 1. There are subsets of the real line for which neither player has a winning strategy in this game. 2. The statement "If X is a set of real numbers, then ONE does not have a winning strategy if, and only if, every countable subset of X is a G_δ-set" is independent of the axioms of classical mathematics. 3. There are spaces whose countable subsets are G_δ-sets, and yet ONE has a winning strategy in this game. 4. For a hereditarily Lindelöf space X, TWO has a winning strategy if, and only if, X is countable. (B) For ℱ the collection of F_σ-subsets of a subset X of the real line the determinacy of this game is independent of ZFC.