Revista Integración, temas de matemáticas.

Revista Integración, temas de matemáticas

Vol. 42 No. 2 (2024): Revista Integración, temas de matemáticas
Accepted articles: Preprint

Some characterizations of the internal structure of Whitney levels

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  • David Maya
David Maya
Universidad Autónoma del Estado de México

Published 2024-11-14

Keywords

  • Aposyndesis,
  • connectedness colocal,
  • decomposability,
  • hyperspace of the subcontinua irreducibility,
  • Whitney level,
  • Wilder continuum
    • ...More
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How to Cite

Maya, D. (2024). Some characterizations of the internal structure of Whitney levels. Revista Integración, Temas De matemáticas, 42(2), 55–65. Retrieved from https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/15823

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Creative Commons License

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Abstract

Let X be a continuum, and let C(X) denote the hyperspace of all subcontinua of X. It is known that there exist monotone maps μ from
C(X) into [0, ∞) such that μ({x}) = 0 for each x ∈ X, and if A is a proper subcontinuum of B, then μ(A) < μ(B). The subcontinua μ−1(t) of C(X) are called Whitney levels of C(X). In this paper, a class of closed subsets of X is employed to characterize the Whitney levels of C(X) possessing one of the following properties: irreducibility, decomposability, being a Wilder continuum, aposyndesis, semiaposyndesis, n-aposyndesis, finite aposyndesis, and connectedness colocal.

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References

  1. Camargo J. and Macías S., “On Wilder, strongly Wilder, continuumwise Wilder, D, D∗, and D∗∗ continua”, Bull. Malays. Math. Sci. Soc., 47 (2024), No. 3, 95-13. doi: 10.1007/s40840-024-01688-2
  2. Camargo J., Maya D. and Pellicer-Covarrubias P., “Noncut subsets of the hyperspace of subcontinua”, Topology Appl., 305 (2022), 107867-18. doi: 10.1016/j.topol.2021.107867
  3. Capulín F., Maya D. and Rios O. I., “Points on the edge of Whitney levels”, preprint.
  4. Espinoza B. and Matsuhashi E., “D-continua, D∗-continua, and Wilder continua”, Topology Appl., 285 (2020), 107393-25. doi: 10.1016/j.topol.2020.107393
  5. Goodykoontz Jr. J. T., “More on connectedness im kleinen and local connectedness in C(X)”, Proc. Amer. Math. Soc., 65 (1977), No. 2, 357-364. doi: 10.2307/2041923
  6. Hughes C. B., “Some properties of Whitney continua in the hyperspace C(X)”, Topology Proceedings, Alabama, USA, 1, 209-219, 1976.
  7. Illanes A. and Nadler Jr. S. B.,Hyperspaces: Fundamentals and Recent Advances, CRC Press, 1999.
  8. Imamura H., Matsuhashi E. and Oshima Y., “Some theorems on decomposable continua”, Topology Appl., 343 (2024), 108794-14. doi: 10.1016/j.topol.2023.108794
  9. Kelley J. L., “Hyperspaces of a continuum”, Trans. Amer. Math. Soc., 52 (1942), 22-36. doi: 10.2307/1990151
  10. Macías S., Set function T . An account on F. B. Jones’ contributions to topology, Springer Nature, vol. 67, 2021. doi: 10.1007/978-3-030-65081-0
  11. Matsuhashi E. and Oshima Y., “Some decomposable continua and Whitney levels of their hyperspaces”, Topology Appl., 326 (2023), 108395-9. doi: 10.1016/j.topol.2022.108395
  12. Rogers, Jr. J. T., “Whitney continua in the hyperspace C(X)”, Pacific J. Math., 58 (1975), No. 2, 569-584. doi: 10.2140/pjm.1975.58.569
  13. Whitney H., “Regular families of curves. I”, Proc. Natl. Acad. Sci. USA, 18 (1932), No. 3, 275-278. doi: 10.1073/pnas.18.3.275
  14. Whitney H., “Regular families of curves. II”, Proc. Natl. Acad. Sci. USA, 18 (1932), No. 4, 340-342. doi: 10.1073/pnas.18.4.340