Playful Boolean algebras

Veličković, Boban

doi:10.1090/S0002-9947-1986-0846604-0

Playful Boolean algebras
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Trans. Amer. Math. Soc. 296 (1986), 727-740 Request permission

Abstract:

We show that for an atomless complete Boolean algebra $\mathcal {B}$ of density $\leq {2^{{\aleph _0}}}$, the Banach-Mazur, the split and choose, and the Ulam game on $\mathcal {B}$ are equivalent. Moreover, one of the players has a winning strategy just in trivial cases: Empty wins iff $\mathcal {B}$ adds a real; Nonempty wins iff $\mathcal {B}$ has a $\sigma$-closed dense set. This extends some previous results of Foreman, Jech, and Vojtáš

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