Glasnik Matematicki, Vol. 55, No. 1 (2020), 113-128.

THE HYPERSPACE OF TOTALLY DISCONNECTED SETS

Raúl Escobedo, Patricia Pellicer-Covarrubias and Vicente Sánchez-Gutiérrez

Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Av. San Claudio y 18 sur, Col. San Manuel, Edificio FM3-210, Ciudad Universitaria C.P. 72570, Puebla, México
e-mail: escobedo@fcfm.buap.mx

Departmento de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, Ciudad de México, C.P. 04510, México
e-mail: paty@ciencias.unam.mx

Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Av. San Claudio y 18 sur, Col. San Manuel, Edificio FM3-210, Ciudad Universitaria C.P. 72570, Puebla, México
e-mail: rompc190787@gmail.com

Abstract.   In this paper we study the hyperspace of all nonempty closed totally disconnected subsets of a space, equipped with the Vietoris topology. We show results of compactness, connectedness and local connectedness for this hyperspace. We also include a study of path connectedness, particularly we prove that for a smooth dendroid this hyperspace is pathwise connected, and we present a general result which implies that for an Euclidean space this hyperspace has uncountably many arc components.

2010 Mathematics Subject Classification.   54B20, 54F15, 54G05

Key words and phrases.   Continuum, hyperspace, locally connected, pathwise connected, totally disconnected set

https://doi.org/10.3336/gm.55.1.10

References:

1. J. J. Charatonik, Two invariants under continuity and the incomparability of fans, Fund. Math. 53 (1963/64), 187-204.
   MathSciNet     CrossRef

2. J. J. Charatonik and C. Eberhart, On smooth dendroids, Fund. Math. 67 (1970), 297-322.
   MathSciNet     CrossRef

3. S. García-Ferreira and Y. F. Otriz-Castillo, The hyperspace of convergent sequences, Topol. Appl. 196 (2015), 795-804.
   MathSciNet     CrossRef

4. S. García-Ferreira and R. Rojas Hernández, Connectedness like properties on the hyperspace of convergent sequences, Topol. Appl. 230 (2017), 639-647.
   MathSciNet     CrossRef

5. A. Illanes and S. B. Nadler, Jr., Hyperspaces. Fundamental and recent advences, Marcel Dekker, Inc., New York, 1999.
   MathSciNet    

6. J. M. Martinez-Montejano, Mutual aposyndesis of symmetric products, Topology Proc. 24 (1999), 203-213.
   MathSciNet    

7. D. Maya, P. Pellicer-Covarrubias and R. Pichardo-Mendoza, General properties of the hyperspace of convergent sequences, Topology Proc. 51 (2018), 143-168.
   MathSciNet    

8. E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182.
   MathSciNet     CrossRef

9. L. Mohler, A characterization of smoothness in dendroids, Fund. Math. 67 (1970), 369-376.
   MathSciNet     CrossRef

10. S. B. Nadler, Jr., Hyperspaces of sets, Marcel Dekker, Inc., New York, 1978.
    MathSciNet    

11. S. B. Nadler, Jr., Continuum theory. An introduction, Marcel Dekker, Inc., New York, 1992.
    MathSciNet    

12. G. T. Whyburn, Analytic topology, Amer. Math. Soc. New York, 1942.
    MathSciNet