Continua whose cone and hyperspace are homeomorphic

Nadler, Sam B.

doi:10.1090/S0002-9947-1977-0464191-0

Continua whose cone and hyperspace are homeomorphic
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Trans. Amer. Math. Soc. 230 (1977), 321-345 Request permission

Abstract:

Let X be a (nonempty) metric continuum. By the hyperspace of X we mean $C(X) = \{ A$: A is a nonempty subcontinuum of $X\}$ with the Hausdorff metric H. It is determined that there are exactly eight hereditarily decomposable continua X such that the cone over X is homeomorphic to $C(X)$. Information about cone-to-hyperspace homeomorphisms, and about arc components for general classes of continua whose cone and hyperspace are homeomorphic is obtained. It is determined that indecomposable continua whose cone and hyperspace are homeomorphic have arcwise connected composants and, if finite-dimensional, have a strong form of the cone = hyperspace property.

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