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.

中文翻译：

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普遍可分离的多样性

摘要 Urysohn 空间是一个可分完备度量空间，具有两个基本性质： (a) 普适性：每个可分度量空间都可以等距嵌入其中；(b) 超均匀性：两个有限子空间之间的每个有限等距都可以扩展到整个空间的自动等距。Urysohn 空间由这两个属性在可分离的度量空间内唯一确定到等距。我们为多样性引入了 Urysohn 空间的类似物，这是最近开发的度量空间概念的变体。在多样性中，任何有限的点集都被分配一个非负值，扩展了仅适用于无序点对的度量概念。我们构建了独特的可分离完全多样性，它对于可分离多样性是超同质和普遍的。