Universal and ultrahomogeneous Polish metric structures

Abstract: We use Fra\" iss\'e theoretic methods to construct several universal and ultrahomogeneous Polish metric structures. Namely, universal and ultrahomogeneous Polish metric space equipped with countably many closed subsets of its powers, universal and ultrahomogeneous Polish metric space equipped with a closed subset of the product of itself and some fixed compact metric space, and universal and ultrahomogeneous Polish metric space equipped with an L-Lipschitz function, for an arbitrary positive L, to some fixed Polish metric space. These results are direct generalization of the classical result of P. Urysohn. Possible applications are discussed.