On completeness of Hausdorff hyperspaces
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916729597198336 |
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| author | Komara, Ján |
| author_facet | Komara, Ján |
| contents | The Hausdorff hyperspace of a metric space consists of all its non-empty bounded closed sets and it is equipped with the Pompeiu--Hausdorff set distance. We present a simpler novel proof that the Hausdorff hyperspace of a complete space is complete as well. The Main Lemma is crucial in this demonstration and though it uses an induction argument -- the only one in our completeness proof -- it is stated purely in terms of neighborhoods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06571 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On completeness of Hausdorff hyperspaces Komara, Ján General Topology 54B20 (Primary) 54E35, 28A80 (Secondary) The Hausdorff hyperspace of a metric space consists of all its non-empty bounded closed sets and it is equipped with the Pompeiu--Hausdorff set distance. We present a simpler novel proof that the Hausdorff hyperspace of a complete space is complete as well. The Main Lemma is crucial in this demonstration and though it uses an induction argument -- the only one in our completeness proof -- it is stated purely in terms of neighborhoods. |
| title | On completeness of Hausdorff hyperspaces |
| topic | General Topology 54B20 (Primary) 54E35, 28A80 (Secondary) |
| url | https://arxiv.org/abs/2505.06571 |