$G_δ$-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 $G_σ$-subsets of a subset X of the real line the determinacy of this game is independent of ZFC.