On generalized homogeneity of locally connected plane continua

Janusz Jerzy Charatonik

Commentationes Mathematicae Universitatis Carolinae (1991)

  • Volume: 32, Issue: 4, page 769-774
  • ISSN: 0010-2628

Abstract

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The well-known result of S. Mazurkiewicz that the simple closed curve is the only nondegenerate locally connected plane homogeneous continuum is extended to generalized homogeneity with respect to some other classes of mappings. Several open problems in the area are posed.

How to cite

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Charatonik, Janusz Jerzy. "On generalized homogeneity of locally connected plane continua." Commentationes Mathematicae Universitatis Carolinae 32.4 (1991): 769-774. <http://eudml.org/doc/247312>.

@article{Charatonik1991,
abstract = {The well-known result of S. Mazurkiewicz that the simple closed curve is the only nondegenerate locally connected plane homogeneous continuum is extended to generalized homogeneity with respect to some other classes of mappings. Several open problems in the area are posed.},
author = {Charatonik, Janusz Jerzy},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {confluent; continuum; dendrite; homogeneous; light; local homeomorphism; locally connected; monotone; open; plane; simple closed curve; universal plane curve; homogeneity; Menger universal curve; Sierpinski universal plane curve},
language = {eng},
number = {4},
pages = {769-774},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On generalized homogeneity of locally connected plane continua},
url = {http://eudml.org/doc/247312},
volume = {32},
year = {1991},
}

TY - JOUR
AU - Charatonik, Janusz Jerzy
TI - On generalized homogeneity of locally connected plane continua
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 4
SP - 769
EP - 774
AB - The well-known result of S. Mazurkiewicz that the simple closed curve is the only nondegenerate locally connected plane homogeneous continuum is extended to generalized homogeneity with respect to some other classes of mappings. Several open problems in the area are posed.
LA - eng
KW - confluent; continuum; dendrite; homogeneous; light; local homeomorphism; locally connected; monotone; open; plane; simple closed curve; universal plane curve; homogeneity; Menger universal curve; Sierpinski universal plane curve
UR - http://eudml.org/doc/247312
ER -

References

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  1. Anderson R.D., A characterization of the universal curve and a proof of its homogeneity, Ann. of Math. (2) 67 (1958), 313-324. (1958) Zbl0083.17607MR0096180
  2. Anderson R.D., One-dimensional continuous curves and a homogeneity theorem, Ann. of Math. (2) 68 (1958), 1-16. (1958) Zbl0083.17608MR0096181
  3. Bing R.H., A simple closed curve is the only homogeneous bounded plane continuum that contains an arc, Canad. J. Math. 12 (1960), 209-230. (1960) Zbl0091.36204MR0111001
  4. Charatonik J.J., On fans, Dissertationes Math. (Rozprawy Mat.) 54 (1967), 1-40. (1967) Zbl0163.44604MR0227944
  5. Charatonik J.J., Some problems on generalized homogeneity of continua, Topology, Proc. International Topological Conference, Leningrad 1982; Lecture Notes in Math., vol. 1060, Springer Verlag, 1984, 1-6. Zbl0544.54029MR0770218
  6. Charatonik J.J., Mappings of the Sierpiński curve onto itself, Proc. Amer. Math. Soc. 92 (1984), 125-132. (1984) Zbl0524.54010MR0749904
  7. Charatonik J.J., Generalized homogeneity of curves and a question of H. Kato, Bull. Polish Acad. Sci. Math. 36 (1988), 409-411. (1988) Zbl0767.54030MR1101430
  8. Charatonik J.J., Monotone mappings of universal dendrites, Topology Appl. 38 (1991), 163-167. (1991) Zbl0726.54012MR1094549
  9. Charatonik J.J., Omiljanowski K., On light open mappings, Proceedings International Topological Conference, Baku (USSR), l989, 211-219. Zbl0817.54010MR1347226
  10. Kato H., Generalized homogeneity of continua and a question of J.J. Charatonik, Houston J. Math. 13 (1987), 51-63. (1987) Zbl0635.54017MR0884233
  11. Kato H., On problems of H. Cook, Topology Appl. 26 (1987), 219-228. (1987) Zbl0612.54042MR0904468
  12. Krasinkiewicz J., On homeomorphisms of the Sierpiński curve, Comment. Math. Prace Mat. 12 (1969), 255-257. (1969) Zbl0235.54039MR0247618
  13. Mazurkiewicz S., Sur les continus homogènes, Fund. Math. 5 (1924), 137-146. (1924) 
  14. Prajs J.R., On open homogeneity of closed balls in the Euclidean spaces, preprint. 
  15. Prajs J.R., Openly homogeneous continua in 2-manifolds, preprint. Zbl0830.54028
  16. Rogers J.T., Jr., Homogeneous continua, Topology Proc. 8 (1983), 213-233. (1983) Zbl0541.54039MR0738476
  17. Rogers J.T., Jr., Classifying homogeneous continua, Proc. of the Topology Symposium at Oxford University (June 1989), preprint. Zbl0776.54023MR1173271
  18. Wilson D.C., Open mappings of the universal curve onto continuous curves, Trans. Amer. Math. Soc. 168 (1972), 497-515. (1972) Zbl0239.54007MR0298630

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