In this paper we show that the Proper Forcing Axiom is preserved under forcing over any poset PP with the following property: In the generalized Banach–Mazur game over PP of length , 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. (shrink)