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Waraszkiewicz spirals revisited

Pavel Pyrih, Benjamin Vejnar (2012)

Fundamenta Mathematicae

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).

Whitney maps-a non-metric case

Janusz Charatonik, Włodzimierz Charatonik (2000)

Colloquium Mathematicae

It is shown that there is no Whitney map on the hyperspace 2 X for non-metrizable Hausdorff compact spaces X. Examples are presented of non-metrizable continua X which admit and ones which do not admit a Whitney map for C(X).

Whitney Preserving Maps onto Dendrites

Eiichi Matsuhashi (2012)

Bulletin of the Polish Academy of Sciences. Mathematics

We prove the following results. (i) Let X be a continuum such that X contains a dense arc component and let D be a dendrite with a closed set of branch points. If f:X → D is a Whitney preserving map, then f is a homeomorphism. (ii) For each dendrite D' with a dense set of branch points there exist a continuum X' containing a dense arc component and a Whitney preserving map f':X' → D' such that f' is not a homeomorphism.

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