{
  "id": "2206.12776",
  "version": "v1",
  "published": "2022-06-26T03:29:37.000Z",
  "updated": "2022-06-26T03:29:37.000Z",
  "title": "Smooth fans that are endpoint rigid",
  "authors": [
    "Rodrigo Hernández-Gutiérrez",
    "Logan C. Hoehn"
  ],
  "comment": "13 pages, 1 figure",
  "categories": [
    "math.GN"
  ],
  "abstract": "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 Erd\\H{o}s space, then $X$ is necessarily homeomorphic to the Lelek fan; this adds to a 1989 result by W{\\l}odzimierz Charatonik.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-26T03:29:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54F50",
      "54F15",
      "54G20",
      "54F65"
    ],
    "keywords": [
      "smooth fan",
      "endpoint rigid",
      "natural numbers",
      "irrational numbers",
      "cantor set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}