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Transactions of the American Mathematical Society

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Growth hyperspaces of Peano continua

Author: D. W. Curtis
Journal: Trans. Amer. Math. Soc. 238 (1978), 271-283
MSC: Primary 54B20; Secondary 54F25, 57N20
MathSciNet review: 482919
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Abstract: For X a nondegenerate Peano continuum, let $ {2^X}$ be the hyperspace of all nonempty closed subsets of X, topologized with the Hausdorff metric. It is known that $ {2^X}$ is homeomorphic to the Hilbert cube. A nonempty closed subspace $ \mathcal{G}$ of $ {2^X}$ is called a growth hyperspace provided it satisfies the following condition: if $ A \in \mathcal{G}$, and $ B \in {2^X}$ such that $ B \supset A$ and each component of B meets A, then also $ B \in \mathcal{G}$. The class of growth hyperspaces includes many previously considered subspaces of $ {2^X}$. It is shown that if X contains no free arcs, and $ \mathcal{G}$ is a nontrivial growth hyperspace, then $ \mathcal{G}\backslash \{ X\} $ is a Hilbert cube manifold. A corollary characterizes those growth hyperspaces which are homeomorphic to the Hilbert cube. Analogous results are obtained for growth hyperspaces with respect to the hyperspace $ {\text{cc}}(X)$ of closed convex subsets of a convex n-cell X.

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Keywords: Hyperspaces, hyperspaces of convex subsets, Peano continuum, Hilbert cube manifold, inverse sequence, near-homeomorphism, local dendron
Article copyright: © Copyright 1978 American Mathematical Society

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