Aronszajn line

An Aronszajn line is a linear ordering of cardinality $aleph\_1$which contains no subset order-isomorphic to
* $omega\_1,$ with the usual ordering
* the reverse of $omega\_1,$
* an uncountable subset of the Real numbers with the usual ordering

Unlike Suslin lines, the existence of Aronszajn lines is provable using the standard axioms of set theory. A linear ordering is an Aronszajn line if and only if it is the lexicographical ordering of some Aronszajn tree. [cite journal
title=Lexicographically ordered trees
last=Funk
first=W.
coauthors=Lutzer, D.
journal=Topology and its Applications
volume=152
date=2005
pages=275–300
doi=10.1016/j.topol.2004.10.011]

References