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Distance Sets of Urysohn Metric Spaces

  Published:2012-07-03
 Printed: Feb 2013
  • N. W. Sauer,
    Department of Mathematics and Statistics, University of Calgary, Calgary, AB
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Abstract

A metric space $\mathrm{M}=(M;\operatorname{d})$ 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 $f$. The space $\mathrm{M}$ is {\em universal} if it isometrically embeds every finite metric space $\mathrm{F}$ with $\operatorname{dist}(\mathrm{F})\subseteq \operatorname{dist}(\mathrm{M})$. (With $\operatorname{dist}(\mathrm{M})$ being the set of distances between points in $\mathrm{M}$.) A metric space $\boldsymbol{U}$ is an {\em Urysohn} metric space if it is homogeneous, universal, separable and complete. (It is not difficult to deduce that an Urysohn metric space $\boldsymbol{U}$ isometrically embeds every separable metric space $\mathrm{M}$ with $\operatorname{dist}(\mathrm{M})\subseteq \operatorname{dist}(\boldsymbol{U})$.) The main results are: (1) A characterization of the sets $\operatorname{dist}(\boldsymbol{U})$ 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 every homogeneous, universal, separable metric space $\mathrm{M}$ is homogeneous.
Keywords: partitions of metric spaces, Ramsey theory, metric geometry, Urysohn metric space, oscillation stability partitions of metric spaces, Ramsey theory, metric geometry, Urysohn metric space, oscillation stability
MSC Classifications: 03E02, 22F05, 05C55, 05D10, 22A05, 51F99 show english descriptions Partition relations
General theory of group and pseudogroup actions {For topological properties of spaces with an action, see 57S20}
Generalized Ramsey theory [See also 05D10]
Ramsey theory [See also 05C55]
Structure of general topological groups
None of the above, but in this section
03E02 - Partition relations
22F05 - General theory of group and pseudogroup actions {For topological properties of spaces with an action, see 57S20}
05C55 - Generalized Ramsey theory [See also 05D10]
05D10 - Ramsey theory [See also 05C55]
22A05 - Structure of general topological groups
51F99 - None of the above, but in this section
 

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