Aronszajn line

In mathematical set theory, an Aronszajn line (named after Nachman Aronszajn) is a linear ordering of cardinality ${\displaystyle \aleph _{1}}$ which contains no subset order-isomorphic to

• ${\displaystyle \omega _{1}}$ with the usual ordering
• the reverse of ${\displaystyle \omega _{1}}$
• 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.[1]

References

1. Funk, Will; Lutzer, David J. (2005). "Lexicographically ordered trees". Topology and Its Applications. 152 (3): 275–300. doi:10.1016/j.topol.2004.10.011. Zbl 1071.03032.