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Layers of components of $ \beta([0,1]\times {\bf N})$ are indecomposable


Author: Michel Smith
Journal: Proc. Amer. Math. Soc. 114 (1992), 1151-1156
MSC: Primary 54D40
DOI: https://doi.org/10.1090/S0002-9939-1992-1093605-4
MathSciNet review: 1093605
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Abstract: We examine the structure of certain subcontinua of the Stone-Čech compactification of the reals. Let $ N$ denote the integers, let $ X = [0,1] \times N$, and let $ C$ be a component of $ {X^*} = \beta X - X$. It is known that $ C$ admits an upper semicontinuous decomposition $ G$ into maximal nowhere dense subcontinua of $ C$ so that $ C/G$ is a Hausdorff arc. The elements of $ G$ are called layers. It has been shown that the layers of $ C$ that contain limit points of a countable increasing or decreasing sequence of cut points of $ C$ are nondegenerate indecomposable continua (various forms of this fact have been proven by Bellamy and Rubin, Mioduszewski, and Smith). We show that all the layers of $ C$ are indecomposable.*


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  • [Be] D. P. Bellamy, A non-metric indecomposable continuum, Duke Math. J. 38 (1971), 15-20. MR 0271911 (42:6792)
  • [B1] A. Blass, Near coherence of filters, II: Applications to operator ideals, the Stone-Čech remainder of a half-line, order ideals of sequences, and slenderness of groups, Trans. Amer. Math. Soc., 300 (1987), no. 2, 557-581. MR 876466 (88d:03094b)
  • [BR] D. P. Bellamy and L. R. Rubin, Indecomposable continua in Stone-Čech compactifications, Proc. Amer. Math. Soc. 39 (1973), 427-432. MR 0315670 (47:4219)
  • [BS] S. Baldwin and M. Smith, On a possible property of far points of $ \beta [0,\infty )$, Topology Proc. 11 (1986), 239-245. MR 945501 (89f:54048)
  • [vD] E. K. van Douwen, The number of subcontinua of the remainder of the plane, Pacific J. Math. 97 (1981), no. 2, 349-355. MR 641164 (83a:54028)
  • [GJ] L. Gillman and M. Jerison, Rings of continuous functions, van Nostrand, Princeton, NJ, 1960. MR 0116199 (22:6994)
  • [J] T. Jech, Set theory, Academic Press, NY, 1968.
  • [K] K. Kuratowski, Topology I & II, Academic Press, NY, 1968.
  • [M] J. Mioduszewski, On composants of $ \beta R - R$, Proc. Conf. Topology and Measure I (Zinnowitz, 1974) (J. Flachsmeyer, Z. Fralik, and F. Terpe, eds.) Ernst-Meritz-Arndt-Universitat zu Grifwald, 1978, pp. 257-283. MR 540576 (80g:54031)
  • [Mo] R. L. Moore, Foundations of point-set theory, Amer. Math. Soc. Colloq. Publ., vol. XII, Amer. Math. Soc., Providence, RI, 1962. MR 0150722 (27:709)
  • [S1] M. Smith, $ \beta [0,\infty )$ does not contain non-degenerate hereditarily indecomposable continua, Proc. Amer. Math. Soc. 101 (1987), 377-384. MR 902559 (88j:54040)
  • [S2] -, $ \beta (X - \{ x\} )$ for $ X$ not locally connected, Topology Appl. 26 (1987), 239-250. MR 904470 (88h:54038)
  • [S3] -, No arbitrary product of $ \beta ([0,\infty )) - [0,\infty )$ contains a nondegenerate hereditarily indecomposable continuum, Topology Appl. 28 (1988), 23-28. MR 927279 (89a:54032)
  • [S4] -, The subcontinua of $ \beta [0,\infty ) - [0,\infty )$, Topology Proc. 11 (1986), 385-413. MR 945509 (89m:54038a)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1992-1093605-4
Keywords: Stone-Čech compactifications, nonmetric continua, indecomposable continua
Article copyright: © Copyright 1992 American Mathematical Society

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