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Concerning continuous selections


Authors: Sam B. Nadler and L. E. Ward
Journal: Proc. Amer. Math. Soc. 25 (1970), 369-374
MSC: Primary 54.55
DOI: https://doi.org/10.1090/S0002-9939-1970-0256360-9
MathSciNet review: 0256360
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Abstract: Necessary and sufficient conditions are given in order that certain types of partially ordered continua admit a continuous selection on the hyperspace of nonempty compact connected subsets. We establish that the class of arcwise connected compacta which admit continuous selections on their space of subcontinua is a proper subclass of the dendroids. This class is also shown to be larger than the class of metrizable generalized trees.


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

  • [1] K. Borsuk and S. Mazurkiewicz, Sur l'hyperspace d'un continu, C. R. Sci. Varsovie 24 (1931), 149-152.
  • [2] C. E. Capel and W. L. Strother, Multi-valued functions and partial order, Portugal. Math. 17 (1958), 41-47. MR 21 #322. MR 0101512 (21:322)
  • [3] J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), 22-36. MR 3, 315. MR 0006505 (3:315b)
  • [4] Virginia Walsh Knight, A continuous partial order for Peano continua, Pacific J. Math. 30 (1969), 141-154. MR 0246266 (39:7570)
  • [5] R. J. Koch, Arcs in partially ordered spaces, Pacific J. Math. 9 (1959), 723-728. MR 21 #7269. MR 0108553 (21:7269)
  • [6] R. J. Koch and I. S. Krule, Weak cutpoint ordering on hereditarily unicoherent continua, Proc. Amer. Math. Soc. 11 (1960), 679-681. MR 22 #11356. MR 0120606 (22:11356)
  • [7] K. Kuratowski, Sam B. Nadler, Jr. and G. S. Young, Continuous selections on locally compact separable metric spaces, Bull. Acad. Polon. Sci. (to appear).
  • [8] M. M. McWaters, Arcs, semigroups, and hyperspaces, Canad. J. Math. 20 (1968), 1207-1210. MR 37 #6914. MR 0231359 (37:6914)
  • [9] E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. MR 13, 54. MR 0042109 (13:54f)
  • [10] L. E. Ward, Jr., Mobs, trees and fixed points, Proc. Amer. Math. Soc. 8 (1957), 798-804. MR 20 #3516. MR 0097036 (20:3516)
  • [11] -, Concerning Koch's theorem on the existence of arcs, Pacific J. Math. 15 (1965), 347-355. MR 31 #6206. MR 0181981 (31:6206)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0256360-9
Keywords: Dendrite, dendroid, directed space, generalized tree, selection
Article copyright: © Copyright 1970 American Mathematical Society

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