Distance Sets of Urysohn Metric Spaces

Sauer, N.W.

Canadian journal of mathematics, Tome 65 (2013) no. 1, pp. 222-240

Résumé

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.

Détail

DOI : 10.4153/CJM-2012-022-4

Mots-clés : 03E02, 22F05, 05C55, 05D10, 22A05, 51F99, partitions of metric spaces, Ramsey theory, metric geometry, Urysohn metric space, oscillation stability