Strongly Baire trees and a cofinal branch principle
A partially ordered set \((P, \leq)\) is said to be \((\omega,\infty)\)-distributive (or Baire) if the intersection of any countable family of dense open subsets of \(P\) is itself a dense open subset of \(P\). The author introduces a stronger property, \(P\) is ``strongly Baire'', which he shows holds if and only if \(P\) is \((\omega,\infty)\)-distributive and forcing with \(P\) preserves stationary subsets of \(\omega_1\). (He also gives a game theoretic characterisation of the strongly Baire property.) He introduces the Cofinal Branch Principle: this is the statement that every strongly Baire tree of height \(\omega_1\) has a cofinal branch. The author observes that the semiproper forcing axiom implies the cofinal branch principle. The main results in this paper are that the cofinal branch principle implies that Todorcevic's strong reflection principle holds, that there is no Souslin tree, and that MA\({}^+\)(\(\sigma\)-closed) holds.
scientific article; zbMATH DE number 1455688
scientific article