On Countably Closed Complete Boolean Algebras

It is unprovable that every complete subalgebra of a countably closed complete Boolean algebra is countably closed. Introduction. A partially ordered set (P,<) is σ-closed if every countable chain in P has a lower bound. A complete Boolean algebra B is countably closed if (B, <) has a dense subset that is σ-closed. In [2] the first author introduced a weaker condition for Boolean algebras, game-closed: the second player has a winning strategy in the infinite game where the two players play an infinite descending chain of nonzero elements, and the second player wins if the chain has a lower bound. In [1], Foreman proved that when B has a dense subset of size א1 and is game-closed then B is countably closed. (By Vojtáš [5] and Veličković [4] this holds for every B that has a dense subset of size 20 .) We show that, in general, it is unprovable that game-closed implies countably closed. We construct a model in which a B exists that is game-closed but not countably closed. It remains open whether a counterexample exists in ZFC. Being game-closed is a hereditary property: If A is a complete subalgebra of a game-closed complete Boolean algebra B then A is game-closed. It is observed in [3] that every game-closed algebra is embedded in a countably closed algebra; in fact, 1991 Mathematics Subject Classification. 03E.