In this paper we consider the special case of the Banach-Mazur game played on a topological space when the space also has an underlying Boolean Algebra structure. This case was first studied by Jech [2]. The version of the Banach-Mazur game we will play is the following game played on the Boolean algebra:

Players I and II alternate moves playing a descending sequence of elements of a Boolean algebra ℬ.

Player II wins the game iff Πi∈ωbi ≠ 0. Jech first considered these games and showed:

Theorem (Jech [2]). ℬ is (ω1, ∞)-distributive iff player I does not have a winning strategy in the game played on ℬ.

If ℬ has a dense ω-closed subset then it is easy to see that player II has a winning strategy in this game. This paper establishes a partial converse to this, namely it gives cardinality conditions on ℬ under which II having a winning strategy implies ω-closure.

In the course of proving the converse, we consider games of length > ω and generalize Jech's theorem to these games. Finally we present an example due to C. Gray that stands in counterpoint to the theorems in this paper.

In this section we give a few basis definitions and explain our notation. These definitions are all standard.