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Distance Sets of Urysohn Metric Spaces

Published online by Cambridge University Press:  20 November 2018

N.W. Sauer
N.W. Sauer*
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, AB, e-mail: nsauer@ucalgary.ca
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Abstract

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A metric space $\text{M=}\left( M;\text{d} \right)$ is homogeneous if for every isometry $f$ of a finite subspace of $\text{M}$ to a subspace of $\text{M}$ there exists an isometry of $\text{M}$ onto $\text{M}$ extending $f$. The space $\text{M}$ is universal if it isometrically embeds every finite metric space $\text{F}$ with $\text{dist}\left( \text{F} \right)\subseteq \text{dist}\left( \text{M} \right)$ (with $\text{dist}\left( \text{M} \right)$ being the set of distances between points in $\text{M}$).

A metric space $U$ is a Urysohn metric space if it is homogeneous, universal, separable, and complete. (We deduce as a corollary that a Urysohn metric space $U$ isometrically embeds every separable metric space $\text{M}$ with $\text{dist}\left( \text{M} \right)\subseteq \text{dist}\left( U \right)$.)

The main results are: (1) A characterization of the sets $\text{dist}\left( U \right)$ for Urysohn metric spaces $U$. (2) If $R$ is the distance set of a Urysohn metric space and $\text{M}$ and $\text{N}$ 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 every homogeneous, universal, separable metric space $\text{M}$ is homogeneous.

Keywords

03E0222F0505C5505D1022A0551F99partitions of metric spacesRamsey theorymetric geometryUrysohn metric spaceoscillation stability
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 1 , 01 February 2013 , pp. 222 - 240
Copyright
Copyright © Canadian Mathematical Society 2013

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