Smooth fans that are endpoint rigid

• Hernández-Gutiérrez, Rodrigo ^[1] ; Hoehn, Logan C. ^[2]
  1. [1] Universidad Autónoma Metropolitana

     Universidad Autónoma Metropolitana

     México

     
  2. [2] Nipissing University

     Nipissing University

     Canadá

     
• Localización: Applied general topology, ISSN-e 1989-4147, ISSN 1576-9402, Vol. 24, Nº. 2, 2023, págs. 407-422
• Idioma: inglés
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• Resumen

  • 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 : X → 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.