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Dendrites

A dendrite is a locally connected continuum containing no simple closed curve. By the order of a point p$ in a dendrite X$, writing \mathrm{ord}\,(p, X)$ \mathrm{ord}\,(p, X)$, we mean the Menger-Urysohn order, see [Kuratowski 1968, §51, I, p. 274], or equivalently, the order in the classical sense, i.e., the number of arcs emanating from p$ and disjoint out of p$ (see [Charatonik 1962, p. 229] and [Lelek 1961, p. 301]).

In this chapter the term of an end point of a continuum is always used in the sense of a point of (Menger-Urysohn) order 1$, i.e., p$ is an end point of X$ provided that \mathrm{ord}\,(p, X) = 1$. For dendrites this concept coincides with one of an end point in the classical sense, but not with the notion of an end point of an arc-like continuum as defined e.g. in [Bing 1951, p. 660].

Given a dendrite X$, we denote by E(X)$ the set of all end points of X$ and by R(X)$ the set of all its ramification points (i.e., points of order at least 3). Various structural as well as mapping characterizations of dendrites are collected in [Charatonik et al. 1998, Theorems 1.1 and 1.2, p. 228 and 230, respectively]. See also [Charatonik et al. 2000].


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Next: Preliminary properties Up: LOCALLY CONNECTED CONTINUA Previous: Graphs
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30