Applications of cohomology to set theory. II: Todorčević trees
type: Item
instance of: scholarly article
title: Applications of cohomology to set theory. II: Todorčević trees
zbMATH Open document ID: 0855.03030
DOI: 10.1016/0168-0072(95)00024-0
author: Daniel E. Talayco
published in: Annals of Pure and Applied Logic
publication date: 1997-02-09T00:00:00Z
review text: The author develops a cohomology theory for a class of \(\omega_1\)-trees. This continues the author's investigations in Part I [ibid. 71, 69-106 (1995; Zbl 0824.03029)] on application of cohomology to gaps. An \(\omega_1\)-tree \(T\) is special if there is a function \(f: T\to \mathbb{Z}\) such that if \(s< t\) then \(f(s)\neq f(t)\). An Aronszajn tree is an \(\omega_1\)-tree with no uncountable chain. It is well known that MA implies that each Aronszajn tree is special. It was observed by Todorčević that special Aronszajn trees can be coded by partitions of pairs of countable ordinals into countably many pieces. This fact is used by the author to apply cohomology to a class of \(\omega_1\)-trees called Todorčević trees. The author shows that there are similarities in the construction of Hausdorff gaps and Todorčević trees. He shows that under \(\text{MA} +\neg \text{CH}\) all Aronszajn trees are special Todorčević trees whereas under \(\diamondsuit\) there are Todorčević trees which are not special Aronszajn trees.
reviewed by: Martin Weese
Mathematics Subject Classification ID: 03E99
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E50
zbMATH DE Number: 908464
zbMATH Keywords: special trees
zbMATH Keywords: diamond
zbMATH Keywords: Martin's Axiom
zbMATH Keywords: cohomology
zbMATH Keywords: \(\omega_ 1\)-trees
zbMATH Keywords: Aronszajn tree
zbMATH Keywords: Todorčević trees
zbMATH Keywords: Hausdorff gaps
MaRDI profile type: Publication
full work available at URL: 0168-0072(95)00024-0
OpenAlex ID: W2061343979
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description: scientific article; zbMATH DE number 908464