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A Hilbert space limit for the iterated hyperspace functor


Authors: H. Toruńczyk and J. West
Journal: Proc. Amer. Math. Soc. 89 (1983), 329-335
MSC: Primary 54B20; Secondary 57N20
DOI: https://doi.org/10.1090/S0002-9939-1983-0712646-1
MathSciNet review: 712646
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Abstract: Let $ X$ be a nondegenerate metric Peano continuum and let $ P(X)$ be the hyperspace of closed, nonvoid subsets of $ X$ equipped with the Hausdorff metric. Then the inclusion of $ X$ into $ P(X)$ as the single element sets is an isometry and we have a direct system $ X \to P(X) \to P(P(X)) \to \cdots $ of isometric inclusions. Let $ X'$ be the metric direct limit and $ {X^*}$ be its completion. We prove that the pair $ ({X^*},X')$ is homeomorphic to the pair $ ({l^2},l_\sigma ^2)$, where $ l_\sigma ^2$ is the linear span in $ {l^2}$ of the Hilbert cube.


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DOI: https://doi.org/10.1090/S0002-9939-1983-0712646-1
Keywords: Hyperspace, direct limit, Peano continuum, Hilbert cube, Hilbert space
Article copyright: © Copyright 1983 American Mathematical Society

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