Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

No homogeneous tree-like continuum contains an arc


Author: Charles L. Hagopian
Journal: Proc. Amer. Math. Soc. 88 (1983), 560-564
MSC: Primary 54F15; Secondary 54F20
DOI: https://doi.org/10.1090/S0002-9939-1983-0699434-X
MathSciNet review: 699434
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In 1960 R. H. Bing [4, p. 210] asked the following question. "Is there a homogeneous tree-like continuum that contains an arc?" We answer this question in the negative. This result generalizes Bing's theorem [4, p. 229] that every atriodic homogeneous tree-like continuum is arcless.


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

  • [1] R. H. Bing, A homogeneous indecomposable plane continuum, Duke Math. J. 15 (1948), 729-742. MR 0027144 (10:261a)
  • [2] -, Snake-like continua, Duke Math. J. 18 (1951), 653-663. MR 0043450 (13:265a)
  • [3] -, Each homogeneous nondegenerate chainable continuum is a pseudo-arc, Proc. Amer. Math. Soc. 10 (1959), 345-346. MR 0105072 (21:3818)
  • [4] -, A simple closed curve is the only homogeneous bounded plane continuum that contains an arc, Canad. J. Math. 12 (1960), 209-230. MR 0111001 (22:1869)
  • [5] -, The elusive fixed point property, Amer. Math. Monthly 76 (1969), 119-132. MR 0236908 (38:5201)
  • [6] C. E. Burgess, Chainable continua and indecomposability, Pacific J. Math. 9 (1959), 653-659. MR 0110999 (22:1867)
  • [7] -, Homogeneous continua which are almost chainable, Canad. J. Math. 13 (1961), 519-528. MR 0126255 (23:A3551)
  • [8] -, Homogeneous $ 1$-dimensional continua, General Topology and Modern Analysis, Academic Press, New York, 1981, pp. 169-175.
  • [9] E. G. Effros, Transformation groups and $ {C^*}$-algebras, Ann. of Math. (2) 81 (1965), 38-55. MR 0174987 (30:5175)
  • [10] C. L. Hagopian, Homogeneous plane continua, Houston J. Math. 1 (1975), 35-41. MR 0383369 (52:4250)
  • [11] -, Indecomposable homogeneous plane continua are hereditarily indecomposable, Trans. Amer. Math. Soc. 224 (1976), 339-350. MR 0420572 (54:8586)
  • [12] -, A characterization of solenoids, Pacific J. Math. 68 (1977), 425-435. MR 0458381 (56:16584)
  • [13] -, Atriodic homogeneous continua, Pacific J. Math. (to appear). MR 749539 (85m:54031)
  • [14] -, F. B. Jones, Certain homogeneous unicoherent indecomposable continua, Proc. Amer. Math. Soc. 2 (1951), 855-859. MR 0045372 (13:573a)
  • [15] I. W. Lewis, Almost chainable homogeneous continua are chainable, Houston J. Math. 7 (1981), 373-377. MR 640978 (83a:54044)
  • [16] -, The pseudo-arc of pseudo-arcs is unique, preprint.
  • [17] R. L. Moore, Concerning triodic continua in the plane, Fund. Math. 13 (1929), 261-263.
  • [18] J. T. Rogers, Jr., Homogeneous, hereditarily indecomposable continua are tree-like, preprint.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54F15, 54F20

Retrieve articles in all journals with MSC: 54F15, 54F20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0699434-X
Keywords: Homogeneity, tree-like continuum, arcless continuum, tree chain, hereditarily indecomposable continuum, pseudo-arc, dog-chases-rabbit principle
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society