Hyperspaces of cones and fans

Eberhart, Carl; Nadler, Sam B.

doi:10.1090/S0002-9939-1979-0542098-5

Hyperspaces of cones and fans
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by Carl Eberhart and Sam B. Nadler

Proc. Amer. Math. Soc. 77 (1979), 279-288

DOI: https://doi.org/10.1090/S0002-9939-1979-0542098-5

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Abstract:

In this paper we investigate the structure of the hyperspaces of subcontinua of some nonlocally connected continua. It is shown that if X is either the cone over any infinite compact metric space or a fan with an infinite number of endpoints, then the space of subcontinua containing the vertex of X is homeomorphic to the Hilbert cube. The hyperspace of subcontinua of a smooth fan (i.e., a subcontinuum of the cone over the Cantor set) is completely described. Also we discuss the question of when two “nicely embedded” copies of the Hilbert cube in a hyperspace have a Hilbert cube sum. In connection with this we describe two spaces, one decomposable and one indecomposable, whose hyperspaces of continua are homeomorphic.

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