Glasnik Matematicki, Vol. 44, No.1 (2009), 195-210.

DENDRITES AND SYMMETRIC PRODUCTS

Gerardo Acosta, Rodrigo Hernández-Gutiérrez and Verónica Martinez-de-la-Vega

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, México D.F., 04510, México
e-mail: gacosta@matem.unam.mx
e-mail: rod@matem.unam.mx
e-mail: vmvm@matem.unam.mx


Abstract.   For a given continuum X and a natural number n, we consider the hyperspace Fn(X) of all nonempty subsets of X with at most n points, metrized by the Hausdorff metric. In this paper we show that if X is a dendrite whose set of end points is closed, n in N and Y is a continuum such that the hyperspaces Fn(X) and Fn(Y) are homeomorphic, then Y is a dendrite whose set of end points is closed.

2000 Mathematics Subject Classification.   54B20, 54C15, 54F15, 54F50.

Key words and phrases.   Continuum, contractibility, dendrite, finite graph, unique hyperspace.


Full text (PDF) (free access)

DOI: 10.3336/gm.44.1.12


References:

  1. G. Acosta, Continua with unique hyperspace, Lecture Notes in Pure and Applied Mathematics 230, Marcel Dekker, Inc., New York, 2002, 33-49.
    MathSciNet

  2. G. Acosta and D. Herrera-Carrasco, Dendrites without unique hyperspace, Houston J. Math. 35 (2009), 469-484.

  3. D. Arévalo, W. J. Charatonik, P. Pellicer-Covarruvias and L. Simón, Dendrites with a closed set of end points, Topology Appl. 115 (2001), 1-17.
    MathSciNet     CrossRef

  4. R. H. Bing, Partitioning a set, Bull. Amer. Math. Soc. 55 (1949), 1101-1110.
    MathSciNet     CrossRef

  5. K. Borsuk, Theory of Retracts, Monografie Matematyczne 44, Polish Scientific Publishers, Warszawa, Poland, 1967.
    MathSciNet

  6. K. Borsuk and S. Ulam, On symmetric products of topological spaces, Bull. Amer. Math. Soc. 37 (1931), 875-882.
    MathSciNet     CrossRef

  7. E. Castaneda and A. Illanes, Finite graphs have unique symmetric products, Topology Appl. 153 (2006), 1434-1450.
    MathSciNet     CrossRef

  8. J. J. Charatonik and W. J. Charatonik, Dendrites, Aportaciones Mat. Comun. 22, Soc. Mat. Mexicana, México (1998), 227--253.
    MathSciNet

  9. J. J. Charatonik and A. Illanes, Local connectedness in hyperspaces, Rocky Mountain J. Math. 36 (2006), 811-856.
    MathSciNet     CrossRef

  10. D. Curtis and N. T. Nhu, Hyperspaces of finite subsets which are homeomorphic to aleph0-dimensional linear metric spaces, Topology Appl. 19 (1985), 251-260.
    MathSciNet     CrossRef

  11. D. Herrera-Carrasco, Dendrites with unique hyperspace, Houston J. Math. 33 (2007), 795-805.
    MathSciNet

  12. D. Herrera-Carrasco and F. Macias-Romero, Dendrites with unique n-fold hyperspace, Topology Proc. 32 (2008), 321-337.

  13. D. Herrera-Carrasco, M. de J. López and F. Macias-Romero, Dendrites with unique symmetric products, to appear in Topology Proc.

  14. W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, Princeton, 1941.
    MathSciNet

  15. A. Illanes, Dendrites with unique hyperspace F2(X), JP J. Geom. Topol. 2 (2002), 75-96.
    MathSciNet

  16. A. Illanes, Dendrites with unique hyperspace C2(X), II, Topology Proc. 34 (2009), 77-96.

  17. A. Illanes and S. B. Nadler Jr., Hyperspaces. Fundamentals and recent advances, Pure and Applied Mathematics 216, Marcel Dekker, Inc., New York, 1999.
    MathSciNet

  18. K. Kuratowski, Topology, Vol. 2, Academic Press and PWN, New York, London and Warszawa, 1968.
    MathSciNet

  19. J. M. Martinez-Montejano, Non-confluence of the natural map of products onto symmetric products, Lecture Notes in Pure and Applied Mathematics 230, Marcel Dekker, Inc., New York, 2002, 229-236.
    MathSciNet

  20. E. E. Moise, Grille decomposition and convexification theorems for compact locally connected continua, Bull. Amer. Math. Soc. 55 (1949), 1111-1121.
    MathSciNet     CrossRef

  21. S. B. Nadler, Jr., Continuum theory. An introduction, Monographs and Textbooks in Pure and Applied Math. 158, Marcel Dekker, Inc., New York, 1992.
    MathSciNet

  22. S. B. Nadler, Jr., Dimension theory: an introduction with exercises, Aportaciones Matemáticas: Textos [Mathematical Contributions: Texts], 18. Sociedad Matemática Mexicana, México, 2002.
    MathSciNet

  23. G. T. Whyburn, Analytic topology, American Mathematical Society Colloquium Publications 28, American Mathematical Society, New York, 1942, reprinted with corrections 1971.
    MathSciNet

  24. W. Wu, Note sur les produits essentiels symétriques des espaces topologiques, C. R. Acad. Sci. Paris 224 (1947), 1139-1141.
    MathSciNet


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