Aronszajn line explained

In mathematical set theory, an Aronszajn line (named after Nachman Aronszajn) 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.^[1]

References

1. Lexicographically ordered trees. Funk . Will. Lutzer . David J.. Topology and Its Applications. 152. 2005. 3. 275–300. 10.1016/j.topol.2004.10.011. 1071.03032.