On isometric universality of spaces of metrics
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866914957912702976 |
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| author | Ishiki, Yoshito Koshino, Katsuhisa |
| author_facet | Ishiki, Yoshito Koshino, Katsuhisa |
| contents | A metric space $(M, d)$ is said to be universal for a class of metric spaces if all metric spaces in the class can be isometrically embedded into $(M, d)$. In this paper, for a metrizable space $Z$ possessing abundant subspaces, we first prove that the space of bounded metrics on $Z$ is universal for all bounded metric spaces (with restricted cardinality). Next, in contrast, we show that if $Z$ is an infinite discrete space, then the space of metrics on $Z$ is universal for all separable metric spaces. As a corollary of our results, if $Z$ is non-compact, or uncountable and compact, then the space of metrics on $Z$ is universal for all compact metric spaces. In addition, if $Z$ is compact and countable, then there exists a compact metric space that can not be isometrically embedded into the space of metrics on $Z$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17701 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On isometric universality of spaces of metrics Ishiki, Yoshito Koshino, Katsuhisa Metric Geometry Functional Analysis General Topology A metric space $(M, d)$ is said to be universal for a class of metric spaces if all metric spaces in the class can be isometrically embedded into $(M, d)$. In this paper, for a metrizable space $Z$ possessing abundant subspaces, we first prove that the space of bounded metrics on $Z$ is universal for all bounded metric spaces (with restricted cardinality). Next, in contrast, we show that if $Z$ is an infinite discrete space, then the space of metrics on $Z$ is universal for all separable metric spaces. As a corollary of our results, if $Z$ is non-compact, or uncountable and compact, then the space of metrics on $Z$ is universal for all compact metric spaces. In addition, if $Z$ is compact and countable, then there exists a compact metric space that can not be isometrically embedded into the space of metrics on $Z$. |
| title | On isometric universality of spaces of metrics |
| topic | Metric Geometry Functional Analysis General Topology |
| url | https://arxiv.org/abs/2409.17701 |