Glasnik Matematicki, Vol. 48, No. 1 (2013), 137-165.

WAŻEWSKI'S UNIVERSAL DENDRITE AS AN INVERSE LIMIT WITH ONE SET-VALUED BONDING FUNCTION

Iztok Banič, Matevž Črepnjak, Matej Merhar, Uroš Milutinović and Tina Sovič

Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, Maribor 2000, Slovenia, and, Institute of Mathematics, Physics and Mechanics, Jadranska 19, Ljubljana 1000, Slovenia
e-mail: iztok.banic@uni-mb.si

Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, Maribor 2000, Slovenia, and, Faculty of Chemistry and Chemical Engineering, University of Maribor, Smetanova 17, Maribor 2000, Slovenia
e-mail: matevz.crepnjak@um.si

Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, Maribor 2000, Slovenia
e-mail: matej.merhar@uni-mb.si

Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška 160, Maribor 2000, Slovenia, and, Institute of Mathematics, Physics and Mechanics, Jadranska 19, Ljubljana 1000, Slovenia
e-mail: uros.milutinovic@uni-mb.si

Faculty of Civil Engineering, University of Maribor, Smetanova 17, Maribor 2000, Slovenia
e-mail: tina.sovic@um.si

Abstract.   We construct a family of upper semi-continuous set-valued functions f:[0,1] → 2[0,1] (belonging to the class of so-called comb functions), such that for each of them the inverse limit of the inverse sequence of intervals [0,1] and f as the only bonding function is homeomorphic to Ważewski's universal dendrite. Among other results we also present a complete characterization of comb functions for which the inverse limits of the above type are dendrites.

2010 Mathematics Subject Classification.   54F50, 54C60.

Key words and phrases.   Continua, inverse limits, upper semi-continuous functions, dendrites, Ważewski's universal dendrite.

Full text (PDF) (free access)

DOI: 10.3336/gm.48.1.12

References:

  1. I. Banič, On dimension of inverse limits with upper semicontinuous set-valued bonding functions, Topology Appl. 154 (2007), 2771-2778.
    MathSciNet     CrossRef

  2. I. Banič, Inverse limits as limits with respect to the Hausdorff metric, Bull. Austral. Math. Soc. 75 (2007), 17-22.
    MathSciNet     CrossRef

  3. I. Banič, Continua with kernels, Houston J. Math. 34 (2008), 145-163.
    MathSciNet    

  4. I. Banič, M. Črepnjak, M. Merhar and U. Milutinović, Limits of inverse limits, Topology Appl. 157 (2010), 439-450.
    MathSciNet     CrossRef

  5. I. Banič, M. Črepnjak, M. Merhar and U. Milutinović, Paths through inverse limits, Topology Appl. 158 (2011), 1099-1112.
    MathSciNet     CrossRef

  6. I. Banič, M. Črepnjak, M. Merhar and U. Milutinović, Towards the complete classification of generalized tent maps inverse limits, Topology Appl. 160 (2013), 63-73.
    MathSciNet     CrossRef

  7. J. J. Charatonik, Monotone mappings of universal dendrites, Topology Appl. 38 (1991), 163-187.
    MathSciNet     CrossRef

  8. W. J. Charatonik and R. P. Roe, Inverse limits of continua having trivial shape, Houston J. Math. 38 (2012), 1307-1312.

  9. A. N. Cornelius, Weak crossovers and inverse limits of set-valued functions, preprint, 2009.

  10. J. Dugundji, Topology, Allyn and Bacon, Inc., Boston, 1966.
    MathSciNet    

  11. S. Greenwood and J. A. Kennedy, Generic generalized inverse limits, Houston J. Math. 38 (2012), 1369-1384.
    MathSciNet    

  12. A. Illanes, A circle is not the generalized inverse limit of a subset of [0, 1]2, Proc. Amer. Math. Soc. 139 (2011), 2987-2993.
    MathSciNet     CrossRef

  13. A. Illanes, S. B. Nadler, Jr., Hyperspaces. Fundamentals and recent advances, Marcel Dekker, Inc., New York, 1999.
    MathSciNet    

  14. W. T. Ingram, W. S. Mahavier, Inverse limits of upper semi-continuous set valued functions, Houston J. Math. 32 (2006), 119-130.
    MathSciNet    

  15. W. T. Ingram, Inverse limits of upper semi-continuous functions that are unions of mappings, Topology Proc. 34 (2009), 17-26.
    MathSciNet    

  16. W. T. Ingram, Inverse limits with upper semi-continuous bonding functions: problems and some partial solutions, Topology Proc. 36 (2010), 353-373.
    MathSciNet    

  17. K. Kuratowski, Topology. Vol. 2, Academic Press, New York-London, Warsaw, 1968.
    MathSciNet    

  18. W. S. Mahavier, Inverse limits with subsets of [0,1] × [0,1], Topology Appl. 141 (2004), 225-231.
    MathSciNet     CrossRef

  19. J. R. Munkres, Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, 1975.
    MathSciNet    

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

  21. V. Nall, Inverse limits with set valued functions, Houston J. Math. 37 (2011), 1323-1332.
    MathSciNet    

  22. V. Nall, Connected inverse limits with set-valued functions, Topology Proc. 40 (2012), 167-177.
    MathSciNet    

  23. V. Nall, Finite graphs that are inverse limits with a set valued function on [0, 1], Topology Appl. 158 (2011), 1226-1233.
    MathSciNet     CrossRef

  24. A. Palaez, Generalized inverse limits, Houston J. Math. 32 (2006), 1107-1119.
    MathSciNet    

  25. S. Varagona, Inverse limits with upper semi-continuous bonding functions and indecomposability, Houston J. Math. 37 (2011), 1017-1034.
    MathSciNet    

  26. T. Ważewski, Sour les courbes de Jordan ne renfermant aucune courbe simple fermée de Jordan, Ann. Soc. Polon. Math. 2 (1923), 49-170.
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