Operations, climbability and the proper forcing axiom

In this paper we show that the Proper Forcing Axiom (PFA) is preserved under forcing over any poset P with the following property: In the generalized Banach-Mazur game over P of length (ω1+1), Player II has a winning strategy which depends only on the current position and the ordinal indicating the number of moves made so far. By the current position we mean: The move just made by Player I for a successor stage, or the infimum of all the moves made so far for a limit stage. As a consequence of this theorem, we introduce a weak form of the square principle and show that it is consistent with PFA.