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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Hyperspaces of cones and fans

Authors: Carl Eberhart and Sam B. Nadler
Journal: Proc. Amer. Math. Soc. 77 (1979), 279-288
MSC: Primary 54B20; Secondary 54F20, 54F50
MathSciNet review: 542098
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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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Keywords: Hyperspace, cone, fan, Hilbert cube, <I>Z</I>-set
Article copyright: © Copyright 1979 American Mathematical Society