Sufficent Conditions for the preservation of Path-Connectedness in an arbitrary metric space
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929326479376384 |
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| author | Andronicou, Savvas Milakis, Emmanouil |
| author_facet | Andronicou, Savvas Milakis, Emmanouil |
| contents | It is proven that if $ (X,d) $ is an arbitrary metric space and $ U $ is a path-connected subset of $ X $ with $M:=\{x_i:\ i\in\{1,2,\dots,k\}\}\subset int(U) $, then the property of path-connectedness is also preserved in the resulting set $ U\setminus M, $ provided that the boundary of each open ball of X is a non-empty and path-connected set. Moreover, under appropriate conditions we extend the above result in the case where the set $ M $ is countably infinite. As a consequence these results maintain path-connectedness for domains with holes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_15871 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sufficent Conditions for the preservation of Path-Connectedness in an arbitrary metric space Andronicou, Savvas Milakis, Emmanouil General Topology It is proven that if $ (X,d) $ is an arbitrary metric space and $ U $ is a path-connected subset of $ X $ with $M:=\{x_i:\ i\in\{1,2,\dots,k\}\}\subset int(U) $, then the property of path-connectedness is also preserved in the resulting set $ U\setminus M, $ provided that the boundary of each open ball of X is a non-empty and path-connected set. Moreover, under appropriate conditions we extend the above result in the case where the set $ M $ is countably infinite. As a consequence these results maintain path-connectedness for domains with holes. |
| title | Sufficent Conditions for the preservation of Path-Connectedness in an arbitrary metric space |
| topic | General Topology |
| url | https://arxiv.org/abs/2404.15871 |