Coarse Proximity and Proximity at Infinity
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2018
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| _version_ | 1866913312209371136 |
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| author | Grzegrzolka, Pawel Siegert, Jeremy |
| author_facet | Grzegrzolka, Pawel Siegert, Jeremy |
| contents | We define coarse proximity structures, which are an analog of small-scale proximity spaces in the large-scale context. We show that metric spaces induce coarse proximity structures, and we construct a natural small-scale proximity structure, called the proximity at infinity, on the set of equivalence classes of unbounded subsets of an unbounded metric space given by the relation of having finite Hausdorff distance. We show that this construction is functorial. Consequently, the proximity isomorphism type of the proximity at infinity of an unbounded metric space $X$ is a coarse invariant of $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1804_10263 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Coarse Proximity and Proximity at Infinity Grzegrzolka, Pawel Siegert, Jeremy Metric Geometry General Topology 54E05, 51F99 We define coarse proximity structures, which are an analog of small-scale proximity spaces in the large-scale context. We show that metric spaces induce coarse proximity structures, and we construct a natural small-scale proximity structure, called the proximity at infinity, on the set of equivalence classes of unbounded subsets of an unbounded metric space given by the relation of having finite Hausdorff distance. We show that this construction is functorial. Consequently, the proximity isomorphism type of the proximity at infinity of an unbounded metric space $X$ is a coarse invariant of $X$. |
| title | Coarse Proximity and Proximity at Infinity |
| topic | Metric Geometry General Topology 54E05, 51F99 |
| url | https://arxiv.org/abs/1804.10263 |