Distance sets of universal and Urysohn metric spaces
Norbert Sauer

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 is {\em homogeneous} if for every isometry of a finite subspace of to a subspace of there exists an isometry of onto extending . A metric space is an {\em Urysohn} metric space if it is homogeneous and separable and complete and if it isometrically embeds every separable metric space with . (With being the set of distances between points in .) The main results are: (1) A characterization of the sets for Urysohn metric spaces . (2) If is the distance set of an Urysohn metric space and and are two metric spaces, of any cardinality with distances in , then they amalgamate disjointly to a metric space with…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Banach Space Theory · Fixed Point Theorems Analysis
