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Zbl 0967.54034
Charatonik, Janusz J.; Charatonik, Włodzimierz J.
Dendrites. (English)
[A] López Mimbela, J. A. (ed.) et al., 30th national congress of the Mexican Mathematical Society, Aguascalientes, México, September 28-October 2, 1997. Proceedings. México: Sociedad Matemática Mexicana. Aportaciones Mat., Comun. 22, 227-253 (1998). ISBN 968-36-7196-9/pbk

A continuum is a nonempty compact connected metric space. A dendrite is a locally connected continuum containing no simple closed curve. This paper is a survey article on dendrites. The authors first compile many known structural and mapping characterizations of dendrites as well as some of their other important properties. The following result is typical:\par Theorem. A continuum is a dendrite if and only if every two distinct points are separated by a third point.\par A dendrite is universal if it contains a homeomorphic image of any other dendrite. The authors then recall known properties of various kinds of universal dendrites. A space $X$ is monotonely homogeneous provided that for every $p,q$ in $X$ there is a surjective monotone mapping $f:X\to X$ such that $f(p)=q$. Next the authors give various sufficient conditions for a dendrite to be monotonely homogeneous and state the following\par Problem: Give any structural characterization of monotonely homogeneous dendrites.\par Next, the authors collect known results about chaotic and rigid dendrites and present some new results on the structure of dendrites with the set of their endpoints closed. Finally, the authors give some open problems in this area.
[T.B.Muenzenberger (Manhattan)]

MSC 2000:: *54F50 Spaces of dimension $\le 1$
54E40 Special maps on metric spaces

Keywords: chaotic dendrite; rigid dendrite; homogeneity; end point; ramification point; universal dendrite; monotonely homogeneous dendrites

Cited in: Zbl 1056.54033

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