${\displaystyle G_{\delta }}$-sets in topological spaces and games

Players ONE and TWO play the following game: In the nth inning ONE chooses a set ${\displaystyle O_n}$ from a prescribed family ℱ of subsets of a space X; TWO responds by choosing an open subset ${\displaystyle T_n}$ of X. The players must obey the rule that ${\displaystyle O_n\subseteq O_{n+1}\subseteq T_{n+1}\subseteq T_n}$ 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 ${\displaystyle G_{\delta }}$-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 ${\displaystyle G_{\delta }}$-set" is independent of the axioms of classical mathematics.   3. There are spaces whose countable subsets are ${\displaystyle G_{\delta }}$-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 ${\displaystyle G_{\sigma }}$-subsets of a subset X of the real line the determinacy of this game is independent of ZFC.