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Transactions of the American Mathematical Society

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The hyperspace of the closed unit interval is a Hilbert cube


Authors: R. M. Schori and J. E. West
Journal: Trans. Amer. Math. Soc. 213 (1975), 217-235
MSC: Primary 54B20; Secondary 57A20
DOI: https://doi.org/10.1090/S0002-9947-1975-0390993-3
MathSciNet review: 0390993
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Abstract: Let X be a compact metric space and let $ {2^X}$ be the space of all nonvoid closed subsets of X topologized with the Hausdorff metric. For the closed unit interval I the authors prove that $ {2^I}$ is homeomorphic to the Hilbert cube $ {I^\infty }$, settling a conjecture of Wojdyslawski that was posed in 1938. The proof utilizes inverse limits and near-homeomorphisms, and uses (and developes) several techniques and theorems in infinite-dimensional topology.


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  • [1] R. D. Anderson, On topological infinite deficiency, Michigan Math. J. 14 (1967), 365-383. MR 35 #4893. MR 0214041 (35:4893)
  • [2] M. Brown, Some applications of an approximate theorem for inverse limits, Proc. Amer. Math. Soc. 11 (1960), 478-483. MR 22 #5959. MR 0115157 (22:5959)
  • [3] D. W. Curtis, Simplicial maps which stabilize to near-homeomorphisms, Compositio Math. 25 (1972), 117-122. MR 47 #5808. MR 0317261 (47:5808)
  • [4] M. K. Fort, Jr. and J. Segal, Minimal representations of the hyperspace of a continuum, Duke Math. J. 32 (1965), 129-137. MR 30 #5281. MR 0175096 (30:5281)
  • [5] N. R. Gray, On the conjecture $ {2^X} \approx {I^\omega }$, Fund. Math. 66 (1969/70), 45-52. MR 41 #1014. MR 0256358 (41:1014)
  • [6] O. H. Keller, Die Homeomorphic der kompakten konvexen Mengen in Hilbertschen Raum, Math. Ann. 105 (1931), 748-758. MR 1512740
  • [7] S. Mazurkiewicz, Sur l'hyperspace d'un continu, Fund. Math. 18 (1932), 171-177. MR 0016652 (8:47g)
  • [8] R.M. Schori, Hyperspace and symmetric products of topological spaces, Fund. Math. 63 (1968), 77-88. MR 38 #661. MR 0232336 (38:661)
  • [9] R. Schori and J. E. West, $ {2^I}$ is homeomorphic to the Hilbert cube, Bull. Amer. Math. Soc. 78 (1972), 402-406. MR 46 #8230. MR 0309119 (46:8230)
  • [10] L. Vietoris, Bereiche Zweiter Ordnung, Monatsh. Math. Phys. 32 (1922), 258-280. MR 1549179
  • [11] T. Wazewski, Sur un continu singulier, Fund. Math. 4 (1923), 214-245.
  • [12] J. E. West, Infinite products which are Hilbert cubes, Trans. Amer. Math. Soc. 150 (1970), 1-25. MR 42 #1055. MR 0266147 (42:1055)
  • [13] -, Mapping cylinders of Hilbert cube factors, General Topology and Appl. 1 (1971), no. 2, 111-125. MR 44 #5984. MR 0288788 (44:5984)
  • [14] M. Wojdyslawski, Sur la contractilité des hyperspaces des continus localement convexes, Fund. Math. 30 (1938), 247-252.
  • [15] -, Retractes absolus et hyperspaces des continus, Fund. Math. 32 (1939), 184-192.
  • [16] E. C. Zeeman, Seminar on combinatorial topology, Institut des Hautes Etudes Scientifiques, Paris, 1963 (mimeographed).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0390993-3
Keywords: Hyperspaces, Hausdorff metric, Hilbert cube, infinite-dimensional topology, inverse limits, near-homeomorphisms, mapping cylinders, Q-factor decompositions
Article copyright: © Copyright 1975 American Mathematical Society

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