Smooth fans that are endpoint rigid

Rodrigo Hernández-Gutiérrez; Logan C. Hoehn

doi:10.4995/agt.2023.17922

Authors

Rodrigo Hernández-Gutiérrez Universidad Autónoma Metropolitana https://orcid.org/0000-0002-5949-0871
Logan C. Hoehn Nipissing University

DOI:

https://doi.org/10.4995/agt.2023.17922

Keywords:

smooth fan, rigidity, Lelek fan, Erdős space, almost zero-dimensional

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 : 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.

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Author Biographies

Rodrigo Hernández-Gutiérrez, Universidad Autónoma Metropolitana

Associate Professor. Department of Mathematics

Logan C. Hoehn, Nipissing University

Associate Professor. Department of Computer Science and Mathematics

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