Strongly Baire trees and a cofinal branch principle
type: Item
instance of: scholarly article
title: Strongly Baire trees and a cofinal branch principle
zbMATH Open document ID: 0951.03042
DOI: 10.1007/BF02773564
published in: Israel Journal of Mathematics
publication date: 2000-07-09T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 03E55
zbMATH DE Number: 1455688
zbMATH Keywords: strongly Baire tree
zbMATH Keywords: Souslin tree
zbMATH Keywords: \((\omega,\infty)\)-distributivity
zbMATH Keywords: strong reflection principle
zbMATH Keywords: semiproper forcing axiom
author: Qi Feng
reviewed by: Neil H. Williams
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1455688