Symmetrically Complete Ordered Sets, Abelian Groups and Fields

We characterize linearly ordered sets, abelian groups and fields that are symmetrically complete, meaning that the intersection over any chain of closed bounded intervals is nonempty. Such ordered abelian groups and fields are important because generalizations of Banach’s Fixed Point Theorem hold in them. We prove that symmetrically complete ordered abelian groups and fields are divisible Hahn products and real closed power series fields, respectively. This gives us a direct route to the construction of symmetrically complete ordered abelian groups and fields, modulo an analogous construction at the level of ordered sets; in particular, this gives an alternative approach to the construction of symmetrically complete fields in [12].