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## The dendrite

In connection with the characterization of dendrites having closed set of their end points (see Property 2 in 1.3.4 above) the classes of those dendrites that do not contain a copy of and those that do not contain a copy of the comb are of some interest (see [Arévalo et al. 2001, p. 8]). The first one is just the class of dendrites with finite orders of ramification points, and it is known to have a universal element according to Property 4 in 1.3.8. The other class has also a universal element . Its construction, given in [Arévalo et al. 2001, p. 9] is the following.

Let be any point in the plane and let be a sequence of points in tending to and such that no three of the points are collinear. Define . Then is homeomorphic to . For every , let be a point in such that for any fixed the sequence tends to , no three of the points are collinear, and that the continuum is a dendrite. We continue constructing dendrites in the same manner, such that we get an increasing sequence of dendrites in having the property that the closure of their union is also a dendrite. Finally define

The following properties of the dendrite are proved in [Arévalo et al. 2001, Theorems 4.5 and 4.6, p. 10].

1. A dendrite is homeomorphic to the dendrite if and only if each of the following conditions is satisfied:
1. contains no copy of ;
2. the set of all end points of is contained in the closure of the set of all ramification points of ;
3. all ramification points of are of order .
2. The dendrite is universal in the class of all dendrites containing no copy of .

See Figure A.

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Next: Self-homeomorphic dendrites Up: Dendrites Previous: Other universal dendrites
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30