The Amalgamation Property and Urysohn Structures in Continuous Logic

In this paper we consider the classes of all continuous L -(pre-)structures for a continuous first-order signature L . We characterize the moduli of continuity for which the classes of finite, countable, or all continuous L -(pre-)structures have the amalgamation property. We also characterize when Urysohn continuous L -(pre)-structures exist, establish that certain classes of finite continuous L -structures are countable Fraïssé classes, prove the coherent EPPA for these classes of finite continuous L -structures, and show that actions by automorphisms on finite L -structures also form a Fraïssé class. As consequences, we have that the automorphism group of the Urysohn continuous L -structure is a universal Polish group and that Hall’s universal locally finite group is contained in the automorphism group of the Urysohn continuous L -structure as a dense subgroup.