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Monotone retracts and some characterizations of dendrites


Authors: G. R. Gordh and Lewis Lum
Journal: Proc. Amer. Math. Soc. 59 (1976), 156-158
MSC: Primary 54F20
DOI: https://doi.org/10.1090/S0002-9939-1976-0423317-X
MathSciNet review: 0423317
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Abstract: Let $ M$ be a metric continuum containing a fixed point $ p$. The following conditions are shown to be equivalent. (i) $ M$ is a dendrite. (ii) Each subcontinuum of $ M$ is a monotone retract of $ M$. (iii) $ M$ is arcwise connected and each subcontinuum of $ M$ containing $ p$ is a monotone retract of $ M$.


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

  • [1] G. R. Gordh Jr., Concerning closed quasi-orders on hereditarily unicoherent continua, Fund. Math. 78 (1973), no. 1, 61–73. MR 0322835
  • [2] John G. Hocking and Gail S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR 0125557
  • [3] Lewis Lum, A characterization of local connectivity in dendroids, Studies in topology (Proc. Conf., Univ. North Carolina, Charlotte, N.C., 1974; dedicated to Math. Sect. Polish Acad. Sci.), Academic Press, New York, 1975, pp. 331–338. MR 0358739
  • [4] -, Order preserving and monotone retracts of a dendroid (submitted).
  • [5] L. E. Rogers, Arcwise connectedness and continuum chainability (submitted).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0423317-X
Keywords: Continuum, dendrite, tree, monotone retract, hereditary unicoherence at $ p$
Article copyright: © Copyright 1976 American Mathematical Society