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Hyperspace retractions for curves

AbstractWe 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...

Atomic mappings and extremal continua.

Janusz J. Charatonik — 1992

Extracta Mathematicae

The notion of atomic mappings was introduced by R. D. Anderson in [1] to describe special decompositions of continua. Soon, atomic mappings turned out to be important tools in continuum theory. In particular, it can be seen in [2] and [5] that these maps are very helpful to construct some special, singular continua. Thus, the mappings have proved to be interesting by themselves, and several of their properties have been discovered, e.g. in [6], [7] and [9]. The reader is referred to Table II of...

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