Dendrites (Q2721242)

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:NEWLINENEWLINENEWLINETheorem. A continuum is a dendrite if and only if every two distinct points are separated by a third point.NEWLINENEWLINENEWLINEA 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 followingNEWLINENEWLINENEWLINEProblem: Give any structural characterization of monotonely homogeneous dendrites.NEWLINENEWLINENEWLINENext, 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.NEWLINENEWLINEFor the entire collection see [Zbl 0948.00025].