G_δ-sets in topological spaces and games

摘要

Players One and Two play the following game: In the nth inning One chooses a set On from a prescribed family T of subsets of a space X Two responds by choosing an open subset Tn of X. The players must obey the rule that On Q On+ C Tn+ C Tn 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 T is a Gj-set. To what extent is the converse true? We show that: