Constructions of Urysohn universal ultrametric spaces
Yoshito Ishiki

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.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Banach Space Theory
