On non-separable components of hyperspaces with the Hausdorff metric

Author R. Cauty
cauty@math.jussieu.fr
Universite Paris 6, Institut de mathematiques de Jussieu

Abstract Let $(X,d)$ be a connected non compact metric space. Suppose the metric $d$ convex and such that every closed bounded subset of $X$ is compact. Let $\mathcal F(X)$ be the space of nonvoid closed subsets of $X$ with the Hausdorff distance associated to $d$. We prove that every component of $\mathcal F(X)$ which contains an unbounded closed subset is homeomorphic to the Hilbert space $\ell^2(2^{\aleph_0})$.
Keywords metric space; Hausdorff distance
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Pages 91-105
Volume 35
Issue 1
Year 2011
Journal Matematychni Studii
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