Aronszajn line

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.