The ⁎-variation of the Banach-Mazur game and forcing axioms

We introduce a property of posets which strengthens (ω1+1)-strategic closedness. This property is defined using a variation of the Banach-Mazur game on posets, where the first player chooses a countable set of conditions instead of a single condition at each turn. We prove PFA is preserved under any forcing over a poset with this property. As an application we reproduce a proof of Magidor's theorem about the consistency of PFA with some weak variations of the square principles. We also argue how different this property is from (ω1+1)-operational closedness, which we introduced in our previous work, by observing which portions of MA+(ω1-closed) are preserved or destroyed under forcing over posets with either property.