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On hyperspaces of polyhedra

Author: Katsuro Sakai
Journal: Proc. Amer. Math. Soc. 110 (1990), 1089-1097
MSC: Primary 57N20; Secondary 54B20
MathSciNet review: 1037223
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Abstract: Let $ Q = {[ - 1,1]^\omega }$ be the Hilbert cube and

$\displaystyle {Q_f} = \left\{ {({x_i}) \in Q\vert{x_i} = 0{\text{except for finitely many }}i} \right\}.$

For a compact connected polyhedron $ X$ with $ \dim X > 0$, the hyperspaces of (nonempty) subcompacta, subcontinua, and compact subpolyhedra of $ X$ are denoted by $ {2^X},C(X)$, and $ {\text{Pol(}}X{\text{)}}$, respectively. And let $ {C^{{\text{Pol}}}}(X) = C(X) \cap {\text{Pol(}}X{\text{)}}$. It is shown that the pair $ ({2^X},{\text{Pol(}}X{\text{)}})$ is homeomorphic to $ (Q,{Q_f})$. In case $ X$ has no free arc, it is also proved that the pair $ (C(X),{C^{{\text{Pol}}}}(X))$ is homeomorphic to $ (Q,{Q_f})$.

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Keywords: Hyperspaces, polyhedron, Hilbert cube, fd-cap set
Article copyright: © Copyright 1990 American Mathematical Society

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