Hausdorff distance between ultrametric balls
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909761079869440 |
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| author | Dovgoshey, Oleksiy |
| author_facet | Dovgoshey, Oleksiy |
| contents | Let $(X, d)$ be an ultrametric space and let $d_H$ be the Hausdorff distance on the set $\bar{\mathbf{B}}_X$ of all closed balls in $(X, d)$. Some interconnections between the properties of the spaces $(X, d)$ and $(\bar{\mathbf{B}}_X, d_H)$ are described. It is established that the space $(\bar{\mathbf{B}}_X, d_H)$ has such properties as discreteness, local finiteness, metrical discreteness, completeness, compactness, local compactness if and only if the space $(X, d)$ has these properties. Necessary and sufficient conditions for the separability of the space $(\bar{\mathbf{B}}_X, d_H)$ are also proved. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_00205 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hausdorff distance between ultrametric balls Dovgoshey, Oleksiy General Topology Primary 54E35, 54E45 Let $(X, d)$ be an ultrametric space and let $d_H$ be the Hausdorff distance on the set $\bar{\mathbf{B}}_X$ of all closed balls in $(X, d)$. Some interconnections between the properties of the spaces $(X, d)$ and $(\bar{\mathbf{B}}_X, d_H)$ are described. It is established that the space $(\bar{\mathbf{B}}_X, d_H)$ has such properties as discreteness, local finiteness, metrical discreteness, completeness, compactness, local compactness if and only if the space $(X, d)$ has these properties. Necessary and sufficient conditions for the separability of the space $(\bar{\mathbf{B}}_X, d_H)$ are also proved. |
| title | Hausdorff distance between ultrametric balls |
| topic | General Topology Primary 54E35, 54E45 |
| url | https://arxiv.org/abs/2509.00205 |