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BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access December 29, 2017

A Universal Separable Diversity

  • David Bryant , André Nies and Paul Tupper EMAIL logo


The Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant of the concept of a metric space. In a diversity any finite set of points is assigned a non-negative value, extending the notion of a metric which only applies to unordered pairs of points.We construct the unique separable complete diversity that it is ultrahomogeneous and universal with respect to separable diversities.

MSC 2010: 51F99; 54E50; 54E99


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Received: 2017-09-06
Accepted: 2017-11-10
Published Online: 2017-12-29
Published in Print: 2017-12-20

© 2018

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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