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Constructions of Urysohn Universal Ultrametric Spaces

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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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Acknowledgments

The author would like to thank Tatsuya Goto for helpful comments. The author would also like to thank the referee for helpful suggestions and valuable comments.

Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

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  1. Photonics Control Technology Team, RIKEN Center for Advanced Photonics, 2-1 Hirasawa, Wako, Saitama 351-0198, Japan

    Yoshito Ishiki

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  1. Yoshito Ishiki
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Correspondence to Yoshito Ishiki.

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Ishiki, Y. Constructions of Urysohn Universal Ultrametric Spaces. P-Adic Num Ultrametr Anal Appl 15, 266–283 (2023). https://doi.org/10.1134/S2070046623040027

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