# Model theory of generalized Urysohn spaces

## Gabriel Conant

### University of Notre Dame

Many well known examples of homogeneous metric spaces and graphs can be viewed as analogs of the rational Urysohn space (for example, the random graph as the Urysohn space with distances {0,1,2}). In this talk, I consider the R-Urysohn space, where R is an arbitrary ordered commutative monoid. I will first construct an extension R* of R, such that any model of the theory of the R-Urysohn space (in a relational language) can be given the structure of an R*-metric space. I will then characterize quantifier elimination in this theory by continuity of addition in R*. Finally, I will characterize various model theoretic properties of the R-Urysohn space (e.g. stability, simplicity, weak elimination of imaginaries) using natural algebraic properties of R.