Displaying similar documents to “A characterisation of a continuous curve”

Waraszkiewicz spirals revisited

Pavel Pyrih, Benjamin Vejnar (2012)

Fundamenta Mathematicae

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We study compactifications of a ray with remainder a simple closed curve. We give necessary and sufficient conditions for the existence of a bijective (resp. surjective) mapping between two such continua. Using those conditions we present a simple proof of the existence of an uncountable family of plane continua no one of which can be continuously mapped onto any other (the first such family, so called Waraszkiewicz's spirals, was created by Z. Waraszkiewicz in the 1930's).

Hyperspace retractions for curves

Charatonik Janusz J., Charatonik Włodzimierz J., Omiljanowski Krzysztof, Prajs Janusz R.

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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...

A fixed-point anomaly in the plane

Charles L. Hagopian, Janusz R. Prajs (2005)

Fundamenta Mathematicae

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We define an unusual continuum M with the fixed-point property in the plane ℝ². There is a disk D in ℝ² such that M ∩ D is an arc and M ∪ D does not have the fixed-point property. This example answers a question of R. H. Bing. The continuum M is a countable union of arcs.