Abstract

In this paper, we give new constructions of Urysohn universal ultrametric spaces. We first characterize a Urysohn universal ultrametric subspace of the space of all continuous functions whose images contain the zero, from a zero-dimensional compact Hausdorff space without isolated points into the space of non-negative real numbers equipped with the nearly discrete topology. As a consequence, the whole function space is Urysohn universal, which can be considered as a non-Archimedean analog of Banach-Mazur theorem. As a more application, we prove that the space of all continuous pseudo-ultrametrics on a zero-dimensional compact Hausdorff space with an accumulation point is a Urysohn universal ultrametric space. This result can be considered as a variant of Wan’s construction of Urysohn universal ultrametric space via the Gromov-Hausdorff ultrametric space.

中文翻译：

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Urysohn 通用超度量空间的构造

摘要

在本文中，我们给出了 Urysohn 通用超度量空间的新构造。我们首先刻画所有图像包含零的连续函数空间的 Urysohn 通用超度量子空间，从没有孤立点的零维紧致 Hausdorff 空间到配备近离散拓扑的非负实数空间。因此，整个函数空间是Urysohn通用的，可以被认为是Banach-Mazur定理的非阿基米德模拟。作为更多的应用，我们证明了具有累加点的零维紧Hausdorff空间上的所有连续伪超度量空间是Urysohn通用超度量空间。这一结果可以被认为是 Wan 通过 Gromov-Hausdorff 超度量空间构造 Urysohn 通用超度量空间的变体。