Smooth fans that are endpoint rigid
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866911748278190080 |
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| author | Hernández-Gutiérrez, Rodrigo Hoehn, Logan C. |
| author_facet | Hernández-Gutiérrez, Rodrigo Hoehn, Logan C. |
| contents | Let $X$ be a smooth fan and denote its set of endpoints by $E(X)$. Let $E$ be one of the following spaces: the natural numbers, the irrational numbers, or the product of the Cantor set with the natural numbers. We prove that there is a smooth fan $X$ such that $E(X)$ is homeomorphic to $E$ and for every homeomorphism $h \colon X \to X$, the restriction of $h$ to $E(X)$ is the identity. On the other hand, we also prove that if $X$ is any smooth fan such that $E(X)$ is homeomorphic to complete Erdős space, then $X$ is necessarily homeomorphic to the Lelek fan; this adds to a 1989 result by Włodzimierz Charatonik. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_12776 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Smooth fans that are endpoint rigid Hernández-Gutiérrez, Rodrigo Hoehn, Logan C. General Topology 54F50 (Primary) 54F15, 54G20, 54F65 (Secondary) Let $X$ be a smooth fan and denote its set of endpoints by $E(X)$. Let $E$ be one of the following spaces: the natural numbers, the irrational numbers, or the product of the Cantor set with the natural numbers. We prove that there is a smooth fan $X$ such that $E(X)$ is homeomorphic to $E$ and for every homeomorphism $h \colon X \to X$, the restriction of $h$ to $E(X)$ is the identity. On the other hand, we also prove that if $X$ is any smooth fan such that $E(X)$ is homeomorphic to complete Erdős space, then $X$ is necessarily homeomorphic to the Lelek fan; this adds to a 1989 result by Włodzimierz Charatonik. |
| title | Smooth fans that are endpoint rigid |
| topic | General Topology 54F50 (Primary) 54F15, 54G20, 54F65 (Secondary) |
| url | https://arxiv.org/abs/2206.12776 |