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## Other universal dendrites

For a given nonempty set we denote by any dendrite satisfying the following two conditions:

(a)
if is a ramification point of , then ;
(b)
for each arc contained in and for every there is in a point with .
It is shown in [Charatonik W.J. et al. 1994, Theorem 6.2, p. 229] that the dendrite is topologically unique, i.e., if two dendrites satisfy conditions (a) and (b) with the same set , then they are homeomorphic. The dendrite is called the standard universal dendrite of orders in . If is a singleton , then is just the standard universal dendrite defined previously.

The following properties of dendrites are known (see [Charatonik W.J. et al. 1994, Theorems 6.4 and 6.6-6.8, p. 230; Corollary 6.10, p. 232]).

1. For any nonempty subset , the dendrite is strongly pointwise self-homeomorphic.
2. If , then the dendrite is universal for the family of all dendrites.
3. If the set is finite with , then is universal for the family of all dendrites having orders of ramification points at most .
4. If the set is infinite and , then is universal for the family of all dendrites having finite orders of ramification points.
5. Nonconstant open images of standard universal dendrites are homeomorphic to if and only if is a nonempty subset of .
6. For any nonempty subset and for an arbitrary dendrite there exists a monotone mapping from onto , [Charatonik et al. 1998, Theorem 2.22, p. 239].
7. For any nonempty subset the dendrite is monotonely homogeneous, [Charatonik 1996, Theorem 3.3, p. 292].

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Next: The dendrite Up: Dendrites Previous: Universal dendrites of order
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30