Skip to content
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

Abstract

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

References

[1] Asuman Guven Aksoy and Zair Ibragimov. Convexity of the Urysohn universal space. Journal of Nonlinear and Convex Analysis, 17(6):1239-1247, 2016.Search in Google Scholar

[2] David Bryant and Paul F. Tupper. Hyperconvexity and tight-span theory for diversities. Advances inMathematics, 231(6):3172 - 3198, 2012.10.1016/j.aim.2012.08.008Search in Google Scholar

[3] David Bryant and Paul F. Tupper. Diversities and the geometry of hypergraphs. Discrete Math. Theor. Comput. Sci., 16(2):1- 20, 2014.10.46298/dmtcs.2080Search in Google Scholar

[4] Christian Delhommé, Claude Laflamme, Maurice Pouzet, and Norbert Sauer. Divisibility of countable metric spaces. European Journal of Combinatorics, 28(6):1746-1769, 2007.10.1016/j.ejc.2006.06.024Search in Google Scholar

[5] Andreas W. M. Dress. Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces. Adv. Math., 53(3):321-402, 1984.Search in Google Scholar

[6] Su Gao. Invariant descriptive set theory, volume 293 of Pure and AppliedMathematics (Boca Raton). CRC Press, Boca Raton, FL, 2009.Search in Google Scholar

[7] Miroslav Katetov. On universal metric spaces. In General topology and its relations to modern analysis and algebra, VI (Prague, 1986), volume 16 of Res. Exp. Math., pages 323-330. Heldermann, Berlin, 1988.Search in Google Scholar

[8] Julien Melleray. Some geometric and dynamical properties of the Urysohn space. Topology and its Applications, 155(14):1531-1560, 2008.10.1016/j.topol.2007.04.029Search in Google Scholar

[9] Andrew Poelstra. On the topological and uniform structure of diversities. Journal of Function Spaces and Applications, 2013:9 pages, 2013.10.1155/2013/675057Search in Google Scholar

[10] Paul Urysohn. Sur un espace métrique universel. Bull. Sci. Math, 51(2):43-64, 1927.Search in Google Scholar

[11] Lionel Nguyen Van Thé. Structural Ramsey theory of metric spaces and topological dynamics of isometry groups. American Mathematical Soc., 2010.10.1090/S0065-9266-10-00586-7Search in Google Scholar

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.

Downloaded on 24.3.2023 from https://www.degruyter.com/document/doi/10.1515/agms-2017-0008/html
Scroll Up Arrow