Universal and ultrahomogeneous Polish metric structures

TL;DR

This paper employs Fraïssé theory to construct various universal and ultrahomogeneous Polish metric structures, extending classical results like Urysohn's space, with potential applications in metric geometry and model theory.

Contribution

It introduces new universal ultrahomogeneous Polish metric structures with additional features, generalizing Urysohn's space using Fraïssé theoretic methods.

Findings

Constructed universal ultrahomogeneous Polish metric spaces with closed subsets.

Extended Urysohn's space to include Lipschitz functions.

Discussed potential applications in related mathematical fields.

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.