Inductive dimensions of coarse proximity spaces
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2019
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| _version_ | 1866909998160805888 |
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| author | Grzegrzolka, Pawel Siegert, Jeremy |
| author_facet | Grzegrzolka, Pawel Siegert, Jeremy |
| contents | In this paper, we generalize Dranishnikov's asymptotic inductive dimension to the setting of coarse proximity spaces. We show that in this more general context, the asymptotic inductive dimension of a coarse proximity space is bigger or equal to the inductive dimension of its boundary, and consequently may be strictly bigger than the covering dimension of the boundary. We also give a condition, called complete traceability, on the boundary of the coarse proximity space under which the asymptotic inductive dimension of a coarse proximity space and the inductive dimension of its boundary coincide. Finally, we show that spaces whose boundaries are $Z$-sets and spaces admitting metrizable compactifications have completely traceable boundaries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1907_09687 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Inductive dimensions of coarse proximity spaces Grzegrzolka, Pawel Siegert, Jeremy General Topology 54E05, 54F45, 54D35, 54D40 In this paper, we generalize Dranishnikov's asymptotic inductive dimension to the setting of coarse proximity spaces. We show that in this more general context, the asymptotic inductive dimension of a coarse proximity space is bigger or equal to the inductive dimension of its boundary, and consequently may be strictly bigger than the covering dimension of the boundary. We also give a condition, called complete traceability, on the boundary of the coarse proximity space under which the asymptotic inductive dimension of a coarse proximity space and the inductive dimension of its boundary coincide. Finally, we show that spaces whose boundaries are $Z$-sets and spaces admitting metrizable compactifications have completely traceable boundaries. |
| title | Inductive dimensions of coarse proximity spaces |
| topic | General Topology 54E05, 54F45, 54D35, 54D40 |
| url | https://arxiv.org/abs/1907.09687 |