Sequential decreasing strong size properties

Féelix Capulín 1 , Miguel A. Lara 1  and Fernando Orozco-Zitli 1
  • 1 Facultad de Ciencias Campus Universitario El Cerrillo, Toluca Estado de México C.P. 50200, Piedras Blancas, México
Féelix Capulín
  • Corresponding author
  • Facultad de Ciencias Campus Universitario El Cerrillo, Universidad Autónoma del Estado de México, Piedras Blancas, Toluca Estado de México C.P. 50200, México
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar
, Miguel A. Lara
  • Facultad de Ciencias Campus Universitario El Cerrillo, Universidad Autónoma del Estado de México, Piedras Blancas, Toluca Estado de México C.P. 50200, México
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar
and Fernando Orozco-Zitli
  • Facultad de Ciencias Campus Universitario El Cerrillo, Universidad Autónoma del Estado de México, Piedras Blancas, Toluca Estado de México C.P. 50200, México
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar

Abstract

Let X be a continuum. The n-fold hyperspace Cn(X), n < ∞, is the space of all nonempty closed subsets of X with at most n components. A topological property P is said to be a (an almost) sequential decreasing strong size property provided that if μ is a strong size map for Cn(X, {tj}j=1 is a sequence in the interval (t,1) such that lim tj = t ∈ [0,1) (t ∈ (0,1)) and each fiber μ−1(tj) has property P, then so does μ−1(t). In this paper we show that the following properties are sequential decreasing strong size properties: being a Kelley continuum, local connectedness, continuum chainability and, unicoherence. Also we prove that indecomposability is an almost sequential decreasing strong size property.

  • [1]

    HOSOKAWA, H.: Induced mappings on hyperspaces, Tsukuba J. Math. 21(1) (1997), 239–250.

  • [2]

    HOSOKAWA, H.: Strong size levels of Cn(X), Houston J. Math. 37(3) (2011), 955–965.

  • [3]

    KELLEY, J. L.: Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), 22–36.

  • [4]

    MACÍAS, S.: Aposyndetic properties of symmetric products of continua, Topology Proc. 22 (1997), 281–296.

  • [5]

    MACÍAS, S.: On symmetric products of continua, Topology Appl. 92 (1999), 173–182.

  • [6]

    MACÍAS, S.: On the hyperspaces Cn(X) of a continuum X, Topology Appl. 109 (2001), 237–256.

  • [7]

    MACÍAS, S.: On the hyperspaces Cn(X) of a continuum X II, Topology Proc. 25 (2000), 255–276.

  • [8]

    MACÍAS, S.—PICENO, C.: Strong size properties, Glas. Mat. Ser. III 48 (68) (2013), 103–114.

  • [9]

    MACÍAS, S.—PICENO, C.: More on strong size properties, Glas. Mat. Ser. III 50 (70) (2015), 467–488.

  • [10]

    NADLER, S. B. Jr.: Hyperspaces of Sets. In: Monographs and Textbooks in Pure and Applied Mathematics 49, Marcel Dekker, New York, Inc., 1978.

  • [11]

    NADLER, S. B. Jr.—WEST, T.: Size levels for arcs, Fund. Math. 109 (1980), 243–255.

  • [12]

    OROZCO-ZITLI, F.: Sequential decreasing Whitney properties. In: Continuum Theory, Lect. Notes Pure Appl. Math. 230, N.Y.: Dekker, 2002, pp. 297–306.

  • [13]

    OROZCO-ZITLI, F.: Sequential decreasing Whitney properties II, Topology Proc. 28(1) (2004), 267–276.

  • [14]

    WHYBURN, G. T.: Analytic Topology. In: Amer. Math. Soc. Collq. Publ. 28, Providence, R. I., 1942.

Purchase article
Get instant unlimited access to the article.
$42.00
Log in
Already have access? Please log in.


Journal + Issues

Mathematica Slovaca, the oldest and best mathematical journal in Slovakia, was founded in 1951 at the Mathematical Institute of the Slovak Academy of Science, Bratislava. It covers practically all mathematical areas. As a respectful international mathematical journal, it publishes only highly nontrivial original articles with complete proofs by assuring a high quality reviewing process.

Search