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Concerning the Cone = Hyperspace Property

Published online by Cambridge University Press:  20 November 2018

Dorothy D. Sherling
Dorothy D. Sherling*
Affiliation:
Smith College, Northampton, Massachusetts
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In this paper it is shown that a sufficient condition for a continuum X to have the cone = hyperspace property is that there exists a selection for C(X)\{X} which, for some Whitney map for C(X), maps each nondegenerate Whitney level homeomorphically onto X. Also, we construct an example of a one-dimensional, nonchainable, noncircle-like continuum which has the cone = hyperspace property. The continuum is described by means of inverse limits using only one bonding map. Each factor space in the inverse limit sequence is the quotient space resulting from an upper semi-continuous decomposition of a disjoint union of simple triods. The bonding map is an adaptation of the bonding map defined by W. T. Ingram in his construction of an atriodic, tree-like continuum which is not chainable [4].

Definitions, notation, and terminology. By continuum we mean a nonempty, compact, connected metric space.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 6 , 01 December 1983 , pp. 1030 - 1048
Copyright
Copyright © Canadian Mathematical Society 1983

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