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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Growth hyperspaces of Peano continua
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by D. W. Curtis
Trans. Amer. Math. Soc. 238 (1978), 271-283
DOI: https://doi.org/10.1090/S0002-9947-1978-0482919-1

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.
References
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Bibliographic Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 238 (1978), 271-283
  • MSC: Primary 54B20; Secondary 54F25, 57N20
  • DOI: https://doi.org/10.1090/S0002-9947-1978-0482919-1
  • MathSciNet review: 482919