A Universal Separable Diversity

TL;DR

This paper introduces a universal, ultrahomogeneous separable diversity space, extending the concept of the Urysohn space from metric spaces to diversities, which assign values to finite point sets.

Contribution

It constructs the first known universal, ultrahomogeneous separable diversity space, generalizing the Urysohn space concept to diversities.

Findings

Constructed the unique separable complete diversity space.

Proved universality and ultrahomogeneity of the space.

Extended the Urysohn space concept to diversities.

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.