Revista Integración, temas de matemáticas.

Revista Integración, temas de matemáticas

Vol. 40 No. 2 (2022): Revista Integración, temas de matemáticas
Research and Innovation Articles

Properties of the (n, m)−fold hyperspace suspension of continua

PDF
  • Gerardo Hernández-Valdez,
  • David Herrera Carrasco,
  • Fernando Macías Romero,
  • Maria de Jesús López
Gerardo Hernández-Valdez
Benemérita Universidad Autónoma de Puebla
David Herrera Carrasco
Benemérita Universidad Autónoma de Puebla
Fernando Macías Romero
Benemérita Universidad Autónoma de Puebla
Maria de Jesús López
Benemérita Universidad Autónoma de Puebla
DOI: https://doi.org/10.18273/revint.v40n2-2022002

Published 2022-09-20

Keywords

  • Aposyndesis,
  • Cantor manifold,
  • Continuum,
  • Colocal connectedness,
  • (n, m)−fold hyperspace suspension,
  • Property (b),
  • Unicoherent
    • ...More
    Less

How to Cite

Hernández-Valdez, G., Herrera Carrasco, D., Macías-Romero, . F., & López, M. de J. (2022). Properties of the (n, m)−fold hyperspace suspension of continua. Revista Integración, Temas De matemáticas, 40(2), 159–168. https://doi.org/10.18273/revint.v40n2-2022002

Copyright (c) 2022 Revista Integración, temas de matemáticas

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

Abstract

Let n, m ∈ N with m ≤ n and X be a metric continuum. We consider the hyperspaces Cn(X) (respectively, Fn(X)) of all nonempty closed subsets of X with at most n components (respectively, n points). The (n, m)−fold hyperspace suspension on X was introduced in 2018 by Anaya, Maya, and Vázquez-Juárez, to be the quotient space Cn(X)/Fm(X) which is obtained from Cn(X) by identifying Fm(X) into a one-point set. In this paper we prove that Cn(X)/Fm(X) contains an n−cell; Cn(X)/Fm(X) has property (b); Cn(X)/Fm(X) is unicoherent; Cn(X)/Fm(X) is colocally connected; Cn(X)/Fm(X) is aposyndetic; and Cn(X)/Fm(X) is finitely aposyndetic

PDF

Downloads

Download data is not yet available.

References

  1. Anaya J.G., Maya D., and Vázquez-Juárez F., “The hyperspace HSn m(X) for a finite graph X is unique”, Topology Appl., 157 (2018), 428–439.
  2. Bennett D.E., “Aposyndetic properties of unicoherent continua”, Pacific J. Math., 37 (1971), no. 3, 585–589.
  3. Curtis D.W., and Nhu N.T., “Hyperspaces of finite subsets which are homeomorphic to ℵ0−dimensional linear metric spaces”, Topology Appl., 19 (1985), 251–260.
  4. Dugundji J., Topology, 2nd ed., BCS Associates, Moscow, Idaho, USA, 1978.
  5. Escobedo R., López M. de J., and Macías S., “On the hyperspace suspension of a continuum”, Topology Appl., 138 (2004), 109–124.
  6. Hernández-Valdez G., Herrera-Carrasco D., López M. de J., and Macías-Romero F.,“Uniqueness of the (n, m)−fold hyperspace suspension for continua”, sent to Topology Appl.
  7. Herrera-Carrasco D., “Dendrites with unique hyperspace”, Houston J. Math., 33 (2007), no. 3, 795–805.
  8. Herrera-Carrasco D., Illanes A., Macías-Romero F., and Vázquez-Juárez F., “Finite graphs have unique hyperspace HSn(X)”, Topology Proc., 44 (2014), 75–95.
  9. Herrera-Carrasco D., López M. de J., and Macías-Romero F., “Framed continua have unique n−fold hyperspace suspension”, Topology Appl., 196 (2015), 652–667.
  10. Herrera-Carrasco D., López M. de J., and Macías-Romero F., “Almost meshed locally connected continua without unique n−fold hyperspace suspension”, Houston J. Math., 44 (2018), no. 4, 1335–1365.
  11. Illanes A. and Nadler, Jr., S.B., Hyperspaces Fundamentals and Recent Advances, Monographs and Textbooks in Pure and Applied Math., vol. 216, Marcel Dekker, Inc., New York, 1999.
  12. Kuratowski K., Topology, Vol. II, Academic Press, New York, 1968.
  13. Levin M., and Sternfeld Y., “The space of subcontinua of a 2–dimensional continuum is infinitely dimensional”, Proc. Amer. Math. Soc., 125 (1997), 2771–2775.
  14. Libreros-López A., Macías-Romero F., and Herrera-Carrasco D., “On the uniqueness of n−fold pseudo-hyperspace suspension for locally connected continua”, Topology Appl., 312 (2022), 108053, 22 pp.
  15. Macías J.C., “On the n−fold pseudo-hyperspace suspensions of continua”, Glas. Mat. Ser. III, 43 (2008), 439–449.
  16. Macías S., “On the hyperspaces Cn(X) of a continuum X”, Topology Appl., 109 (2001), 237–256.
  17. Macías S., “On the n−fold hyperspace suspension of continua”, Topology Appl., 138 (2004), 125–138.
  18. Macías S., “On the n−fold hyperspace suspension of continua, II”, Glas. Mat. Ser. III, 41 (2006), no. 61, 335–343.
  19. Macías S., Topics on continua, 2nd ed., Springer, Cham, Switzerland, 2018.
  20. Macías S., and Nadler, Jr. S.B., “n−fold hyperspace, cones, and products”, Topology Proc., 26 (2001–2002), 255–270.
  21. Montero-Rodríguez G., Herrera-Carrasco D., López M. de J., and Macías-Romero F., “Finite graphs have unique n−fold symmetric product suspension”, Houston J. Math., 47 (2021), no. 4, 20 pp.
  22. Morales-Fuentes U., “Finite graphs have unique n−fold pseudo-hyperspace suspension”, Topology Proc., 52 (2018), 219–233.
  23. Nadler, Jr., S.B., Hyperspaces of sets, Monographs and Textbooks in Pure and Applied Math. vol. 49, Marcel Dekker, Inc., New York, 1978.
  24. Nadler, Jr., S.B., “A fixed point theorem for hyperspace suspensions”, Houston J. Math., 5 (1979), 125–132.
  25. Nadler, Jr., S.B., Continuum Theory: An Introduction, Monographs and Textbooks in Pure and Applied Math., vol. 158, Marcel Dekker, Inc., New York, 1992.
  26. Nadler, Jr., S.B., Dimension Theory: An introduction with exercises, Aportaciones Matemáticas Serie Textos 18, Sociedad Matemática Mexicana, Mexico, 2002.
  27. Whyburn G.T., Analytic Topology, Amer. Math. Soc. Colloq. Publ., vol. 28, American Mathematical Society, Providence, RI, 1942.