On Some Model Theoretic Properties of Totally Bounded Ultrametric Spaces

Abstract

Continuing investigations initiated by the first author, we associate relational structures for metric spaces and investigate their model theoretic properties. In this paper, we consider ultrametric spaces. Among others, we show that any elementary substructure of the relational structure associated with a totally bounded ultrametric space X is dense in X. Further, we provide an explicit upper bound for a splitting chain of atomic types in ultrametric spaces of a finite spectrum. For ultrametric spaces, these results improve previous ones of the present authors and may have further practical applications in designing similarity detecting algorithms.