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 $O_n\subseteq O_{n+1}\subseteq T_{n+1}\subseteq T_n$ 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 $G_{\delta }$-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...