Constructions of Urysohn universal ultrametric spaces

TL;DR

This paper introduces new methods for constructing Urysohn universal ultrametric spaces, characterizing subspaces within function spaces and demonstrating their universality, extending classical theorems to non-Archimedean contexts.

Contribution

It provides novel constructions and characterizations of Urysohn universal ultrametric spaces, including a non-Archimedean analog of the Banach–Mazur theorem and variants of Wan's construction.

Findings

The entire function space is Urysohn universal.

The space of continuous pseudo-ultrametrics is a Urysohn universal ultrametric space.

New constructions extend classical results to non-Archimedean settings.

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.