Winning the pressing down game but not Banach Mazur

Abstract

Let $S$ be the set of those $α\inω_2$ that have cofinality $ω_1$. It is consistent relative to a measurable that the nonempty player wins the pressing down game of length $ω_1$, but not the Banach Mazur game of length $ω+1$ (both games starting with $S$).

New version: some improvements in presentation, references, history; main theorem slightly strengthened; some typos removed