Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Hyperspaces homeomorphic to Hilbert space

Author: D. W. Curtis
Journal: Proc. Amer. Math. Soc. 75 (1979), 126-130
MSC: Primary 54B20; Secondary 54F65
MathSciNet review: 529228
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The hyperspace $ {2^X}$ of a metric space X is the space of nonempty compact subsets, topologized by the Hausdorff metric. It is shown that $ {2^X}$ is homeomorphic to the separable Hilbert space $ {l^2}$ if and only if X is connected, locally connected, separable, topologically complete, and nowhere locally compact. The principal tool in the proof is Torunczyk's mapping characterization of $ {l^2}$.

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

  • [1] R. D. Anderson, Hilbert space is homeomorphic to the countable infinite product of lines, Bull. Amer. Math. Soc. 72 (1966), 515-519. MR 0190888 (32:8298)
  • [2] D. W. Curtis, Hyperspaces of noncompact metric spaces, Compositio Math. (to appear). MR 563538 (81c:54009)
  • [3] D. W. Curtis and R. M. Schori, $ {2^X}$ and $ C(X)$ are homeomorphic to the Hilbert cube, Bull. Amer. Math. Soc. 80 (1974), 927-931. MR 0353235 (50:5719)
  • [4] -, Hyperspaces of Peano continua are Hilbert cubes, Fund. Math. 101 (1978), 19-38. MR 512241 (80k:54010)
  • [5] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 0193606 (33:1824)
  • [6] N. Kroonenberg, Pseudo-interiors of hyperspaces, Compositio Math. 32 (1976), 113-131. MR 0413109 (54:1230)
  • [7] U. Tasmetov, On the connectedness of hyperspaces, Soviet Math. Dokl. 15 (1974), 502-504. MR 0341375 (49:6126)
  • [8] H. Torunczyk, Characterizing Hilbert space topology (preprint). MR 611763 (82i:57016)
  • [9] M. Wojdyslawski, Retractes absolus et hyperespaces des continus, Fund. Math. 32 (1939), 184-192.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54B20, 54F65

Retrieve articles in all journals with MSC: 54B20, 54F65

Additional Information

Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society