Connected components of groups and o-minimal expansions of real closed fields

Real closed fields are ordered fields where every positive element is a square, and every polynomial of odd degree has a root. Equivalently, an ordered field F is real closed if and only if the field extension F ( √−1) is algebraically closed. The main example of a real closed field is the field of real numbers. It is wellknown that a real closed field has a unique ordering, so any isomorphism of real closed fields is order-preserving. In [5], it is proved that any two real closed fields whose order types are the same uncountable saturated ordering are isomorphic. Recall than an ηα set is a dense linear ordering without endpoints of power אα in which any cut given by fewer than אα formulas is realized. The ηα sets were introduced by Hausdorff in [7]. Theorem (Erdös-Gillman-Henriksen [5]) Any two real closed fields whose underlying order types are ηα sets are isomorphic. 1 Email: a.conversano@massey.ac.nz 2 Email: pillay@maths.leeds.ac.uk Available online at www.sciencedirect.com