Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Continua whose cone and hyperspace are homeomorphic


Author: Sam B. Nadler
Journal: Trans. Amer. Math. Soc. 230 (1977), 321-345
MSC: Primary 54F20
DOI: https://doi.org/10.1090/S0002-9947-1977-0464191-0
MathSciNet review: 0464191
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let X be a (nonempty) metric continuum. By the hyperspace of X we mean $ C(X) = \{ A$: A is a nonempty subcontinuum of $ X\} $ with the Hausdorff metric H. It is determined that there are exactly eight hereditarily decomposable continua X such that the cone over X is homeomorphic to $ C(X)$. Information about cone-to-hyperspace homeomorphisms, and about arc components for general classes of continua whose cone and hyperspace are homeomorphic is obtained. It is determined that indecomposable continua whose cone and hyperspace are homeomorphic have arcwise connected composants and, if finite-dimensional, have a strong form of the cone = hyperspace property.


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

  • [1] R. H. Bing, Snake-like continua, Duke Math. J. 18 (1951), 653-663. MR 13, 265. MR 0043450 (13:265a)
  • [2] K. Borsuk and S. Mazurkiewicz, Sur l'hyperespace d'un continu, C. R. Soc. Sci. Warsaw 24 (1931), 149-152.
  • [3] G. R. Gordh, Jr. and Sam B. Nadler, Jr., Arc components of chainable Hausdorff continua. General Topology and Appl. 3 (1973), 63-76. MR 47 #5847. MR 0317300 (47:5847)
  • [4] W. Hurewicz and H. Wallman, Dimension theory, Princeton Univ. Press, Princeton, N.J., 1948. MR 3, 312. MR 0006493 (3:312b)
  • [5] J. L. Kelley, Hyperspaces of a continuum, Trans. Amer. Math. Soc. 52 (1942), 22-36. MR 3, 315. MR 0006505 (3:315b)
  • [6] K. Kuratowski, Topology, Vol. II, English transl., Academic Press, New York; PWN, Warsaw, 1968. MR 41 #4467. MR 0259835 (41:4467)
  • [7] Sam B. Nadler, Jr., Arc components of certain chainable continua, Canad. Math. Bull. 14 (1971), 183-189. MR 46 #9949. MR 0310851 (46:9949)
  • [8] -, Continua which are a one-to-one continuous image of $ [0,\infty )$, Fund. Math. 75 (1972), 123-133. MR 47 #5848. MR 0317301 (47:5848)
  • [9] -, Continua whose cones and hyperspaces are homeomorphic, Notices Amer. Math. Soc. 19 (1972), A718-A719. Abstract #72T-G150.
  • [10] -, Locating cones and Hilbert cubes in hyperspaces, Fund. Math. 79 (1973), 233-250. MR 48 #12449. MR 0334130 (48:12449)
  • [11] -, Multicoherence techniques applied to inverse limits, Trans. Amer. Math. Soc. 157 (1971), 227-234. MR 43 #5482. MR 0279761 (43:5482)
  • [12] Sam B. Nadler, Jr. and J. Quinn, Embeddability and structure properties of real curves, Mem. Amer. Math. Soc. No. 125 (1972). MR 50 #5762. MR 0353278 (50:5762)
  • [13] -, Embedding certain compactifications of a half-ray, Fund. Math. 78 (1973), 217-225. MR 47 #9544. MR 0321011 (47:9544)
  • [14] James T. Rogers, Jr., The cone=hyperspace property, Canad. J. Math. 24 (1972), 279-285. MR 45 #4370. MR 0295302 (45:4370)
  • [15] -, Continua with cones homeomorphic to hyperspaces, General Topology and Appl. 3 (1973), 283-289. MR 50 #14699. MR 0362257 (50:14699)
  • [16] -, Dimension of hyperspaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), 177-179. MR 51 #6762. MR 0370535 (51:6762)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54F20

Retrieve articles in all journals with MSC: 54F20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0464191-0
Keywords: Chainable, circle-like, compactification, composant, decomposable continuum, dimension, Hausdorff metric, indecomposable continuum, remainder of a compactification, segment (in the sense of Kelley)
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society