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On atriodic tree-like continua


Authors: Lex G. Oversteegen and E. D. Tymchatyn
Journal: Proc. Amer. Math. Soc. 83 (1981), 201-204
MSC: Primary 54F20; Secondary 54B20, 54F50, 54H25
DOI: https://doi.org/10.1090/S0002-9939-1981-0620013-2
MathSciNet review: 620013
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Abstract: D. P. Bellamy has recently shown that atriodic tree-like continua do not have the fixed point property for homeomorphisms. J. B. Fugate and T. B. McLean showed that hereditarily indecomposable tree-like continua have the fixed point property for pointwise periodic homeomorphisms. In this paper the latter result is extended to the case of atriodic tree-like continua. In the course of the proof it is shown that the property of being an atriodic tree-like continuum is a Whitney property. In particular, it is shown that the hyperspace of an atriodic tree-like continuum is at most $ 2$-dimensional.


References [Enhancements On Off] (What's this?)

  • [1] D. P. Bellamy, A tree-like continuum without the fixed point property, Houston J. Math. 6 (1980), 1-13. MR 575909 (81h:54039)
  • [2] J. H. Case and R. E. Chamberlin, Characterizations of tree-like continua, Pacific J. Math. 10 (1960), 73-84. MR 0111000 (22:1868)
  • [3] J. B. Fugate and T. B. McLean, Compact groups of homeomorphisms on tree-like continua (to appear). MR 626493 (82m:54036)
  • [4] J. Grispolakis and E. D. Tymchatyn, Continua which are images of weakly confluent mappings only. I, Houston J. Math. 5 (1979), 483-502. MR 567908 (81d:54008)
  • [5] -, On confluent mappings and essential mappings, Rocky Mountain J. Math. (to appear).
  • [6] J. Krasinkiewicz, On the hyperspaces of hereditarily indecomposable continua, Fund. Math. 84 (1974), 175-186. MR 0350711 (50:3203)
  • [7] -, On the hyperspaces of snake-like and circle-like continua, Fund. Math. 83 (1974), 155-164. MR 0418058 (54:6102)
  • [8] S. B. Nadler, Jr., Whitney-reversible properties, Fund. Math. (to appear). MR 597070 (82b:54044)
  • [9] -, Hyperspaces of sets, Dekker, New York, 1978. MR 0500811 (58:18330)
  • [10] A. R. Pears, Dimension theory of general spaces, Cambridge Univ. Press, Cambridge, 1975. MR 0394604 (52:15405)
  • [11] J. T. Rogers, Jr., Applications of a Vietoris-Begle theorem for multi-valued maps to the cohomology of hyperspaces, Michigan Math. J. 22 (1976), 315-319. MR 0397678 (53:1536)
  • [12] R. B. Sher, Realizing cell-like maps in Euclidean space, General Topology Appl. 2 (1972), 75-89. MR 0303546 (46:2683)
  • [13] R. H. Sorgenfrey, Concerning triodic continua, Amer. J. Math. 66 (1944), 439-460. MR 0010968 (6:96d)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0620013-2
Keywords: Atriodic tree-like continua, pointwise periodic homeomorphism, fixed points, Whitney property
Article copyright: © Copyright 1981 American Mathematical Society

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