Growth hyperspaces of Peano continua

Author:
D. W. Curtis

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

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Abstract: For *X* a nondegenerate Peano continuum, let be the hyperspace of all nonempty closed subsets of *X*, topologized with the Hausdorff metric. It is known that is homeomorphic to the Hilbert cube. A nonempty closed subspace of is called a *growth hyperspace* provided it satisfies the following condition: if , and such that and each component of *B* meets *A*, then also . The class of growth hyperspaces includes many previously considered subspaces of . It is shown that if *X* contains no free arcs, and is a nontrivial growth hyperspace, then 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 of closed convex subsets of a convex *n*-cell *X*.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1978-0482919-1

Keywords:
Hyperspaces,
hyperspaces of convex subsets,
Peano continuum,
Hilbert cube manifold,
inverse sequence,
near-homeomorphism,
local dendron

Article copyright:
© Copyright 1978
American Mathematical Society