Hyperspace Retractions for Curves


We study retractions from the hyperspace of all nonempty closed subsets of a given continuum onto the continuum (which is naturally embedded in the hyperspace). Some necessary and some sufficient conditions for the existence of such a retraction are found if the continuum is a curve. It is shown that the existence of such a retraction for a curve implies that the curve is a uniformly arcwise connected dendroid, and that a universal smooth dendroid admits such a retraction. The existence of this retraction for a given dendroid implies that the dendroid admits a mean. An example of a (nonplanable) smooth dendroid that admits no mean is constructed. Some related results are obtained and open problems are stated. The results answer several questions asked in the literature.


Mathematics and Statistics

Keywords and Phrases

arc-smooth; curve; dendrite; dendroid; hyperspace; inverse limit; mean; retraction; selection; smooth; universal element

Document Type

Article - Journal

Document Version


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© 1997 Institute of Mathematics, Polish Academy of Sciences, All rights reserved.

Publication Date

01 Jan 1997

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