A short proof of nonhomogeneity of the pseudo-circle
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- by Krystyna Kuperberg and Kevin Gammon
- Proc. Amer. Math. Soc. 137 (2009), 1149-1152
- DOI: https://doi.org/10.1090/S0002-9939-08-09605-6
- Published electronically: September 17, 2008
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Abstract:
The pseudo-circle is known to be nonhomogeneous. The original proofs of this fact were discovered independently by L. Fearnley and J. T. Rogers, Jr. The purpose of this paper is to provide an alternative, very short proof based on a result of D. Bellamy and W. Lewis.References
- David P. Bellamy and Wayne Lewis, An orientation reversing homeomorphism of the plane with invariant pseudo-arc, Proc. Amer. Math. Soc. 114 (1992), no. 4, 1145–1149. MR 1092915, DOI 10.1090/S0002-9939-1992-1092915-4
- R. H. Bing, A homogeneous indecomposable plane continuum, Duke Math. J. 15 (1948), 729–742. MR 27144
- R. H. Bing, Concerning hereditarily indecomposable continua, Pacific J. Math. 1 (1951), 43–51. MR 43451
- R. H. Bing, Each homogeneous nondegenerate chainable continuum is a pseudo-arc, Proc. Amer. Math. Soc. 10 (1959), 345–346. MR 105072, DOI 10.1090/S0002-9939-1959-0105072-6
- Lawrence Fearnley, The pseudo-circle is unique, Bull. Amer. Math. Soc. 75 (1969), 398–401. MR 246265, DOI 10.1090/S0002-9904-1969-12193-2
- Lawrence Fearnley, The pseudo-circle is not homogeneous, Bull. Amer. Math. Soc. 75 (1969), 554–558. MR 242126, DOI 10.1090/S0002-9904-1969-12241-X
- Lawrence Fearnley, The pseudo-circle is unique, Trans. Amer. Math. Soc. 149 (1970), 45–64. MR 261559, DOI 10.1090/S0002-9947-1970-0261559-6
- Charles L. Hagopian, The fixed-point property for almost chainable homogeneous continua, Illinois J. Math. 20 (1976), no. 4, 650–652. MR 418057
- Sze-tsen Hu, Homotopy theory, Pure and Applied Mathematics, Vol. VIII, Academic Press, New York-London, 1959. MR 0106454
- Judy Kennedy and James T. Rogers Jr., Orbits of the pseudocircle, Trans. Amer. Math. Soc. 296 (1986), no. 1, 327–340. MR 837815, DOI 10.1090/S0002-9947-1986-0837815-9
- B. Knaster, Un continu dont tout sous-continu est indécomposable, Fund. Math. 3 (1922), 247-286.
- K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1968. New edition, revised and augmented; Translated from the French by A. Kirkor. MR 0259835
- Wayne Lewis, Almost chainable homogeneous continua are chainable, Houston J. Math. 7 (1981), no. 3, 373–377. MR 640978
- Wayne Lewis, The pseudo-arc, Continuum theory and dynamical systems (Arcata, CA, 1989) Contemp. Math., vol. 117, Amer. Math. Soc., Providence, RI, 1991, pp. 103–123. MR 1112808, DOI 10.1090/conm/117/1112808
- Wayne Lewis, The pseudo-arc, Bol. Soc. Mat. Mexicana (3) 5 (1999), no. 1, 25–77. MR 1692467
- Edwin E. Moise, An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua, Trans. Amer. Math. Soc. 63 (1948), 581–594. MR 25733, DOI 10.1090/S0002-9947-1948-0025733-4
- James T. Rogers Jr., The pseudo-circle is not homogeneous, Trans. Amer. Math. Soc. 148 (1970), 417–428. MR 256362, DOI 10.1090/S0002-9947-1970-0256362-7
- James T. Rogers Jr., Homogeneous, separating plane continua are decomposable, Michigan Math. J. 28 (1981), no. 3, 317–322. MR 629364
Bibliographic Information
- Krystyna Kuperberg
- Affiliation: Department of Mathematics and Statistics, Auburn University, Auburn, Alabama 36849
- Email: kuperkm@auburn.edu
- Kevin Gammon
- Affiliation: Department of Mathematics and Statistics, Auburn University, Auburn, Alabama 36849
- Email: gammokb@auburn.edu
- Received by editor(s): March 7, 2008
- Published electronically: September 17, 2008
- Communicated by: Alexander N. Dranishnikov
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 137 (2009), 1149-1152
- MSC (2000): Primary 54F15, 54F50
- DOI: https://doi.org/10.1090/S0002-9939-08-09605-6
- MathSciNet review: 2457457
Dedicated: Dedicated to James T. Rogers, Jr., on the occasion of his 65th birthday