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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
DOI: https://doi.org/10.1090/S0002-9939-1979-0529228-6
MathSciNet review: 529228
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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}$.


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DOI: https://doi.org/10.1090/S0002-9939-1979-0529228-6
Article copyright: © Copyright 1979 American Mathematical Society