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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.10080 |
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| _version_ | 1866915546454294528 |
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| author | Kim, Donghan |
| author_facet | Kim, Donghan |
| contents | We construct a multiset space $\mathbb{N}[X]$ over a metric space $X$ that simultaneously enjoys desirable topological properties and admits a natural matching metric $d_{\mathbb{N}[X]}$, making it a metrizable abelian topological monoid whose structure is compatible with the original metric on $X$. This framework extends naturally to the free abelian group $\mathbb{Z}[X]$, where a metric $d_{\mathbb{Z}[X]}$ induces a metrizable abelian topological group structure. We further identify the metric completion of $\mathbb{N}[X]$, showing that it carries a canonical extension of the matching metric. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10080 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Metric Topologies on Multiset Spaces as Topological Monoids and Their Group Completion Kim, Donghan Metric Geometry Primary 54E35, Secondary 54H11 We construct a multiset space $\mathbb{N}[X]$ over a metric space $X$ that simultaneously enjoys desirable topological properties and admits a natural matching metric $d_{\mathbb{N}[X]}$, making it a metrizable abelian topological monoid whose structure is compatible with the original metric on $X$. This framework extends naturally to the free abelian group $\mathbb{Z}[X]$, where a metric $d_{\mathbb{Z}[X]}$ induces a metrizable abelian topological group structure. We further identify the metric completion of $\mathbb{N}[X]$, showing that it carries a canonical extension of the matching metric. |
| title | Metric Topologies on Multiset Spaces as Topological Monoids and Their Group Completion |
| topic | Metric Geometry Primary 54E35, Secondary 54H11 |
| url | https://arxiv.org/abs/2510.10080 |