An uncountable family of copies of a non-chainable tree-like continuum in the plane
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Abstract:
A well-known theorem of R. L. Moore states that the plane does not contain an uncountable family of pairwise disjoint triods. In 1974, Ingram demonstrated that the same is not true for non-chainable tree-like continua. The continua in Ingram’s family are not pairwise homeomorphic, making the example less applicable to the study of homogeneous continua in the plane. In this paper, we construct a non-chainable tree-like continuum $X$ such that the product of $X$ with the Cantor set can be embedded in the plane.References
- Howard Cook, W. T. Ingram, and Andrew Lelek, A list of problems known as Houston problem book, Continua (Cincinnati, OH, 1994) Lecture Notes in Pure and Appl. Math., vol. 170, Dekker, New York, 1995, pp. 365–398. MR 1326857, DOI 10.2307/3618329
- Edward G. Effros, Transformation groups and $C^{\ast }$-algebras, Ann. of Math. (2) 81 (1965), 38–55. MR 174987, DOI 10.2307/1970381
- Charles L. Hagopian, Indecomposable homogeneous plane continua are hereditarily indecomposable, Trans. Amer. Math. Soc. 224 (1976), no. 2, 339–350 (1977). MR 420572, DOI 10.1090/S0002-9947-1976-0420572-1
- L. C. Hoehn, A non-chainable plane continuum with span zero, Fund. Math. 211 (2011), no. 2, 149–174. MR 2747040, DOI 10.4064/fm211-2-3
- W. T. Ingram, An uncountable collection of mutually exclusive planar atriodic tree-like continua with positive span, Fund. Math. 85 (1974), no. 1, 73–78. MR 355999, DOI 10.4064/fm-85-1-73-78
- W. T. Ingram, Hereditarily indecomposable tree-like continua, Fund. Math. 103 (1979), no. 1, 61–64. MR 535836, DOI 10.4064/fm-103-1-61-64
- F. Burton Jones, On a certain type of homogeneous plane continuum, Proc. Amer. Math. Soc. 6 (1955), 735–740. MR 71761, DOI 10.1090/S0002-9939-1955-0071761-1
- B. Knaster and C. Kuratowski, Problème 2, Fund. Math. 1 (1920).
- R. L. Moore, Concerning triods in the plane and the junction points of plane continua, Proc. Natl. Acad. Sci. USA 14 (1928), no. 1, 85–88.
- Sam B. Nadler Jr., Continuum theory, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker, Inc., New York, 1992. An introduction. MR 1192552
- Lex G. Oversteegen and E. D. Tymchatyn, Plane strips and the span of continua. I, Houston J. Math. 8 (1982), no. 1, 129–142. MR 666153
- Lex G. Oversteegen and E. D. Tymchatyn, On span and chainable continua, Fund. Math. 123 (1984), no. 2, 137–149. MR 755627, DOI 10.4064/fm-123-2-137-149
- S. Todorčević, Embeddability of $K\times C$ into $X$, Bull. Cl. Sci. Math. Nat. Sci. Math. 22 (1997), 27–35. MR 1612441
- Gerald S. Ungar, On all kinds of homogeneous spaces, Trans. Amer. Math. Soc. 212 (1975), 393–400. MR 385825, DOI 10.1090/S0002-9947-1975-0385825-3
- Eric K. van Douwen, Uncountably many pairwise disjoint copies of one metrizable compactum in another, Topology Appl. 51 (1993), no. 2, 87–91. MR 1229705, DOI 10.1016/0166-8641(93)90142-Z
Additional Information
- L. C. Hoehn
- Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
- Address at time of publication: Department of Computer Science and Mathematics, Nipissing University, 100 College Drive, Box 5002, North Bay, Ontario, Canada P1B 8L7
- MR Author ID: 854228
- Email: lhoehn@uab.edu, loganh@nipissingu.ca
- Received by editor(s): October 11, 2011
- Published electronically: March 4, 2013
- Communicated by: Alexander N. Dranishnikov
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 2543-2556
- MSC (2010): Primary 54F15, 54F50
- DOI: https://doi.org/10.1090/S0002-9939-2013-11760-0
- MathSciNet review: 3043034