Distance sets of universal and Urysohn metric spaces

TL;DR

This paper characterizes the distance sets of Urysohn metric spaces, explores their amalgamation properties, and examines the homogeneity of completions of certain metric spaces, advancing understanding of universal homogeneous metric spaces.

Contribution

It provides a characterization of the distance sets for Urysohn spaces, shows their amalgamation capabilities, and analyzes homogeneity preservation in completions.

Findings

Characterization of distance sets for Urysohn spaces.

Amalgamation of metric spaces with distances in a Urysohn space's set.

Homogeneity of completions of certain homogeneous metric spaces.

Abstract

A metric space $\mathrm{M}=(M;\de)$ is {\em homogeneous} if for every isometry $f$ of a finite subspace of $\mathrm{M}$ to a subspace of $\mathrm{M}$ there exists an isometry of $\mathrm{M}$ onto $\mathrm{M}$ extending $f$. A metric space $\boldsymbol{U}$ is an {\em Urysohn} metric space if it is homogeneous and separable and complete and if it isometrically embeds every separable metric space $\mathrm{M}$ with $\dist(\mathrm{M})\subseteq \dist(\boldsymbol{U})$. (With $\dist(\mathrm{M})$ being the set of distances between points in $\mathrm{M}$.) The main results are: (1) A characterization of the sets $\dist(\boldsymbol{U})$ for Urysohn metric spaces $\boldsymbol{U}$. (2) If $R$ is the distance set of an Urysohn metric space and $\mathrm{M}$ and $\mathrm{N}$ are two metric spaces, of any cardinality with distances in $R$, then they amalgamate disjointly to a metric space with distances in $R$. (3) The completion of a homogeneous separable metric space $\mathrm{M}$ which embeds isometrically every finite metric space $\mathrm{F}$ with $\dist(\mathrm{F})\subseteq \dist(\mathrm{M})$ is homogeneous.