Distance Sets of Urysohn Metric Spaces

作者：Sauer N W^*

来源：Canadian Journal of Mathematics, 2013, 65(1): 222-240.

DOI：10.4153/CJM-2012-022-4

• partitions of metric spaces
• Ramsey theory
• metric geometry
• Urysohn metric space
• oscillation stability

A metric space M = (M; d) is homogeneous if for every isometry f of a finite subspace of M to a subspace of M there exists an isometry of M onto M extending f. The space M is universal if it isometrically embeds every finite metric space F with dist(F) subset of dist(M) (with dist(M) being the set of distances between points in M). %26lt;br%26gt;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 M with dist(M) subset of dist(U).) %26lt;br%26gt;The main results are: (1) A characterization of the sets dist(U) for Urysohn metric spaces U. (2) If R is the distance set of a Urysohn metric space and M and 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 M is homogeneous.

• 出版日期2013-2