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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0390993-3
PII: S 0002-9947(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