Skip to main content
SpringerLink
Log in

Inverse limits of arcs and of simple closed curves with confluent bonding mappings

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

It is proved that Knaster's type continua and solenoids can be considered as inverse limits of arcs and of circles with confluent bonding mappings. Several other classes of bonding mappings, which are relative to confluent ones, also are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. H. Bing, Snake-like continua,Duke Math. J. 18 (1951), 653–663.MR 13, 265

    Google Scholar 

  2. R. H. Bing, A simple closed curve is the only homogeneous bounded plane continuum that contains an arc,Canad. J. Math. 12 (1960), 209–230.MR 22: 1869

    Google Scholar 

  3. M. Brown, Some applications of an approximation theorem for inverse limits,Proc. Amer. Math. Soc. 11 (1960), 478–483.MR 22: 5959

    Google Scholar 

  4. C. E. Capel, Inverse limit spaces,Duke Math. J. 21 (1954), 233–245.MR 15, 976

    Google Scholar 

  5. J. J. Charatonik, Confluent mappings and unicoherence of continua,Fund. Math. 56 (1964), 213–220.MR 31: 723

    Google Scholar 

  6. J. J. Charatonik, Generalized homogeneity and some characterizations of solenoids,Bull. Pol. Acad. Sci. Math. 31 (1983), 171–174.MR 85h: 54021

    Google Scholar 

  7. J. J. Charatonik andW. J. Charatonik, Monotoneity relative to a point and inverse limits of continua,Glasnik Mat. 20 (40) (1985), 139–151.

    Google Scholar 

  8. J. J. Charatonik andW. J. Charatonik, On projections and limit mappings of inverse systems of compact spaces,Topology Appl. 16 (1983), 1–9.MR 85b: 54016

    Google Scholar 

  9. W. J. Charatonik, Inverse limits of smooth continua,Comment. Math. Univ. Carolin. 23, 1 (1982), 183–191.MR 83d: 54054

    Google Scholar 

  10. C. Chevalley,Theory of Lie groups, Princeton University Press, 1946.MR 7, 412

  11. D. van Dantzig, Über homogene Kontinua,Fund. Math. 15 (1930), 102–125.

    Google Scholar 

  12. D. van Dantzig,Topologisch-Algebraische Verkenning, Zeven voordrachten over Topologie, Gerinchem, 1950, pp. 56–79.MR 12, 348

  13. W. Dębski,A topological classification of indecomposable continua of Knaster's type and their composants, Silesian University, Katowice (Poland), 1981. (Ph. D. thesis, in Polish)

    Google Scholar 

  14. W. Dębski, On topological types of the simplest indecomposable continua,Colloq. Math. 49 (1985), 203–211.

    Google Scholar 

  15. S. Eilenberg, Sur quelques propriétés des transformations localement homéomorphes,Fund. Math. 24 (1935), 35–42.Zbl 10, 182

    Google Scholar 

  16. S. Eilenberg andN. Steenrod,Foundations of algebraic topology, Princeton University Press, 1952.MR 14, 398

  17. R. Engelking,General topology, PWN — Polish Scientific Publishers, Warszawa, 1977.MR 58: 18316

    Google Scholar 

  18. R. Engelking andA. Lelek, Metrizability and weight of inverses under confluent mappings,Colloq. Math. 21 (1970), 239–246.MR 41: 7646

    Google Scholar 

  19. B. B. Epps, Jr., Strongly confluent mappings,Notices Amer. Math. Soc. 19 (1972), Abstract 699-G16, A-807.

  20. B. B. Epps, Jr. A classification of continua and confluent transformations, University of Houston, Houston, Texas, U.S.A., 1973. (Ph. D. thesis)

    Google Scholar 

  21. J. N. Grispolakis, On a problem of Lelek concerning open mappings,Colloq. Math. 39 (1978), 253–261.MR 80c: 54010

    Google Scholar 

  22. J. Grispolakis andE. D. Tychatyn, On confluent mappings and essential mappings — a survey,Rocky Mountain J. Math. 11 (1981) 131–153.MR 82k: 54055

    Google Scholar 

  23. C. L. Hagopian, A characterization of solenoids,Pacific J. Math. 68 (1977), 425–435.MR 56: 16584

    Google Scholar 

  24. J. Jobe, Dendrites, dimension, and the inverse arc function,Pacific J. Math. 45 (1973), 245–256.MR 47: 7715

    Google Scholar 

  25. P. Krupski, The property of Kelley in circularly chainable and in chainable continua,Bull. Acad. Polon. Sci., Ser. Sci. Math. 29 (1981), 377–381.MR 83a: 54043

    Google Scholar 

  26. P. Krupski, Solenoids and inverse limits of sequences of arcs with open bonding maps,Fund. Math. 120 (1984), 41–52.MR 85f: 54071

    Google Scholar 

  27. K. Kuratowski,Topology, vol. 2, Acad. Press and PWN, 1968.MR 41: 4467

  28. A. Lelek, Sur l'unicohérence, les homéomorphies locales et les continus irréductibles,Fund. Math. 45 (1957), 51–63.MR 20: 6076

    Google Scholar 

  29. A. Lelek, A classification of mappings pertinent to curve theory,Proc. Univ. Oklahoma Topology Conference, Norman, Oklahoma, 1972; 97–103.MR 50: 11126

    Google Scholar 

  30. A. Lelek, The OM-mappings of continua,Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys. 25 (1977), 781–785.MR 58: 24169

    Google Scholar 

  31. A. Lelek andJ. Mycielski, Some conditions for a mapping to be a covering,Fund. Math. 49 (1961), 295–300.MR 30: 5278

    Google Scholar 

  32. A. Lelek andD. R. Read, Compositions of confluent mappings and some other classes of functions,Colloq. Math. 29 (1974), 101–112.MR 51: 4142

    Google Scholar 

  33. A. Lelek andE. D. Tymchatyn, Pseudo-confluent mappings and a classification of continua,Canad. J. Math. 27 (1975), 1336–1348.MR 54: 6066

    Google Scholar 

  34. T. Maćkowiak, Semi-confluent mappings and their invariants,Fund. Math. 79 (1973), 251–264.MR 47: 9577

    Google Scholar 

  35. T. Maćkowiak, Continuous mappings on continua,Dissertationes Math. (Rozprawy Matematyczne)158 (1979), 1–91.MR 81a: 54034

    Google Scholar 

  36. S. Mardesič andJ. Segal,ε-mappings onto polyhedra,Trans. Amer. Math. Soc. 109 (1963), 146–164.MR 28: 1592

    Google Scholar 

  37. M. C. McCord, Inverse limit sequences with covering maps,Trans. Amer. Math. Soc. 114 (1965), 197–209.MR 30: 3450

    Google Scholar 

  38. S. B. Nadler, Jr., Concerning completeness of the space of confluent mappings,Houston J. Math. 2 (1976), 561–580.MR 55: 9014

    Google Scholar 

  39. S. B. Nadler, Jr. Hyperspaces of sets, Dekker, New York, 1978.MR 58: 18330

    Google Scholar 

  40. D. R. Read, Confluent and related mappings,Colloq. Math. 29 (1974), 233–239.MR 51: 4145

    Google Scholar 

  41. J. W. Rogers, Jr., On mapping indecomposable continua onto certain chainable indecomposable continua,Proc. Amer. Math. Soc. 25 (1970), 449–456.MR 41: 1017

    Google Scholar 

  42. E. H. Spanier,Algebraic topology, McGraw-Hill, New York, 1966.MR 35: 1007

    Google Scholar 

  43. N. Steenrod,The topology of fibre boundles, Princeton University Press, 1951.MR 12, 522

  44. L. Vietoris, Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen,Math. Ann. 97 (1927), 454–472.

    Google Scholar 

  45. A. D. Wallace, Quasi-monotone transformations,Duke Math. J. 7 (1940), 136–145.MR 2, 179

    Google Scholar 

  46. R. W. Wardle, On a property of J. L. Kelley,Houston J. Math. 3 (1977), 291–299.MR 56: 16582

    Google Scholar 

  47. W. T. Watkins,Homeomorphic classification of certain inverse limit spaces with open bonding maps, University of Wyoming, Laramie, Wyoming, U.S.A., 1980. (Ph. D. thesis)

    Google Scholar 

  48. G. T. Whyburn, Open mappings on locally compact spaces,Mem. Amer. Math. Soc. 1 (1950), 1–24.MR 13, 764

    Google Scholar 

  49. G. T. Whyburn, Analytic topology,Amer. Math. Soc. Colloq. Publ. 28, 1963.MR 32: 425

Download references

Author information

Authors and Affiliations

  1. Instytut Matematyczny Uniwersytet, Pl. Grunwaldzki 2/4, PL-50384, Wrocław, Poland

    J. J. Charatonik

Authors
  1. J. J. Charatonik
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Charatonik, J.J. Inverse limits of arcs and of simple closed curves with confluent bonding mappings. Period Math Hung 16, 219–236 (1985). https://doi.org/10.1007/BF01848072

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01848072

AMS (MOS) subject classifications (1980)

Key words and phrases

Search

Navigation