Boundaries of coarse proximity spaces and boundaries of compactifications
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2018
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| _version_ | 1866913312231391232 |
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| author | Grzegrzolka, Pawel Siegert, Jeremy |
| author_facet | Grzegrzolka, Pawel Siegert, Jeremy |
| contents | In this paper, we introduce the boundary $\mathcal{U}X$ of a coarse proximity space $(X,\mathcal{B},{\bf b}).$ This boundary is a subset of the boundary of a certain Smirnov compactification. We show that $\mathcal{U}X$ is compact and Hausdorff and that every compactification of a locally compact Hausdorff space induces a coarse proximity structure whose corresponding boundary is the boundary of the compactification. We then show that many boundaries of well-known compactifications arise as boundaries of coarse proximity spaces. In particular, we give four coarse proximity structures whose boundaries are the Gromov, visual, Higson, and Freudenthal boundaries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1812_09802 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Boundaries of coarse proximity spaces and boundaries of compactifications Grzegrzolka, Pawel Siegert, Jeremy General Topology Metric Geometry 54E05, 54D35 In this paper, we introduce the boundary $\mathcal{U}X$ of a coarse proximity space $(X,\mathcal{B},{\bf b}).$ This boundary is a subset of the boundary of a certain Smirnov compactification. We show that $\mathcal{U}X$ is compact and Hausdorff and that every compactification of a locally compact Hausdorff space induces a coarse proximity structure whose corresponding boundary is the boundary of the compactification. We then show that many boundaries of well-known compactifications arise as boundaries of coarse proximity spaces. In particular, we give four coarse proximity structures whose boundaries are the Gromov, visual, Higson, and Freudenthal boundaries. |
| title | Boundaries of coarse proximity spaces and boundaries of compactifications |
| topic | General Topology Metric Geometry 54E05, 54D35 |
| url | https://arxiv.org/abs/1812.09802 |