A universal Aronszajn line (Q1012971)

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.