Modified: Nov 26, 2011
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 (\omega_1 + 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.[Back]
Kada proved in a previous paper (Topology Appl., 2009) that the collection of compatible metrics on a locally compact separable metriz- able space has the same cofinal type, in the sense of Tukey relation, as the set of functions from \omega to \omega with respect to eventually dominating order. By generalizing this result, we characterize the order structure of the collection of compatible metrics on a separable metrizable space in terms of generalized Galois?Tukey connection.[Back]
We show that large fragments of MM, e.g. the tree property and stationary reflection, are preserved by strongly (\omega_1+1)-game closed forcings. PFA can be destroyed by a strongly (\omega_1+1)-game closed forcing but not by an \omega_2-closed.[Back]
We introduce properties of Boolean algebras which are closely related to the existence of winning strategies in the Banach-Mazur Boolean game. A \sigma-short Boolean algebra is a Boolean algebra that has a dense subset in which every strictly descending sequence of length \omega does not have a nonzero lower bound. We give a characterization of \sigma-short Boolean algebras and study properties of \sigma-short Boolean algebras.[Back]
We show that for any infinite cardinal \kappa, every strongly (\kappa+1)-strategically closed poset is strongly (\kappa^+)-strategically closed if and only if AP_\kappa (the approachability property) holds, answering the question asked in [5]. We also give a complete classification of strengths of strategic closure properties and that of strong strategic closure properties respectively.[Back]
We show that the existence of a precipitous ideal over the successor of some limit cardinals implies the existence of some large cardinals, in the sense of consistency. Moreover we use the same technique to evaluate the consistency strength of precipitousness of Woodin's stationary tower.[Back]
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