A universal Aronszajn line
type: Item
instance of: scholarly article
title: A universal Aronszajn line
zbMATH Open document ID: 1179.03055
author: Justin Tatch Moore
published in: Mathematical Research Letters
publication date: 2009-04-28T00:00:00Z
review text: An Aronszajn line is an uncountable linear order which has no uncountable separable suborders and does not contain a copy of \(\omega_1\) or \(- \;\omega_1\). A Countryman line is an uncountable linear order whose Cartesian square can be covered by countably many non decreasing relations. Fix a Countryman line \(C\), and let \(\zeta_C\) denote the direct sum \(- \;C \oplus \{ 0\} \oplus C\). The following three results are proved : (1) (PFA) Any Aronszajn line is isomorphic to a suborder of the Aronszajn line \(\eta_C\), where \(\eta_C\) is the subset of the lexicographic power \((\zeta_C)^\omega\) consisting of those elements which are eventually \(0\). (2) (PFA) Let \(A\) be an Aronszajn line which is not Countryman. Then \(A\) contains an isomorphic copy of both \(C\) and \(- \;C\). (3) (MA\(_{\aleph_1}\)) Any two \(\aleph_1\)-dense non-stationary Countryman lines are either isomorphic or reverse isomorphic.
reviewed by: Pierre Matet
Mathematics Subject Classification ID: 03E75
Mathematics Subject Classification ID: 06A05
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E35
zbMATH DE Number: 5548670
zbMATH Keywords: PFA
zbMATH Keywords: Aronszajn line
zbMATH Keywords: uncountable linear order
zbMATH Keywords: Countryman line
MaRDI profile type: Publication
DOI: 10.4310/MRL.2009.V16.N1.A12
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description: scientific article; zbMATH DE number 5548670