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Path connectedness, local path connectedness and contractibility of $\mathcal{S}_c(X)$

Volume 160 / 2020

Javier Camargo, David Maya, Patricia Pellicer-Covarrubias Colloquium Mathematicum 160 (2020), 183-211 MSC: 54A20, 54B20. DOI: 10.4064/cm7516-1-2019 Published online: 24 January 2020

Abstract

The hyperspace of all nontrivial convergent sequences in a Hausdorff space $X$ is denoted by $\mathcal {S}_c(X)$. This hyperspace is endowed with the Vietoris topology. In connection with a question and a problem by García-Ferreira, Ortiz-Castillo and Rojas-Hernández, we study the path connectedness and contractibility of $\mathcal {S}_c(X)$. We present necessary conditions on $X$ for the path connectedness of $\mathcal {S}_c(X)$, and also some sufficient conditions. Further, we characterize the local path connectedness of $\mathcal {S}_c(X)$ in terms of that of $X$. We prove the contractibility of $\mathcal {S}_c(X)$ for a class of spaces, and finally we study the connectedness of Whitney blocks and Whitney levels for $\mathcal {S}_c(X)$.

Authors

  • Javier CamargoEscuela de Matemáticas
    Facultad de Ciencias
    Universidad Industrial de Santander
    Ciudad Universitaria, Carrera 27 Calle 9
    Bucaramanga, Santander, A.A. 678, Colombia
    e-mail
  • David MayaUniversidad Autónoma del Estado de México
    Facultad de Ciencias
    Instituto Literario 100, Col. Centro
    Toluca, CP 50000, Mexico
    e-mail
    e-mail
  • Patricia Pellicer-CovarrubiasDepartamento de Matemáticas
    Facultad de Ciencias
    Circuito ext. s/n
    Ciudad Universitaria, C.P. 04510
    CDMX, Mexico
    e-mail

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