## 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
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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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## 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