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## Universal dendrites of order

Let . By the standard universal dendrite of order we mean a dendrite such that each ramification point of is of order and for each arc the set of ramification points of which belong to is dense in . Their constructions, known again from [Wazewski 1923, Chapter K, p. 137] (see also [Charatonik 1991, (4), p. 168]; for the inverse limit construction see [Chaaratonik 1980, p. 491]), mimic that of the Wazewski universal dendrite , but instead of copies of we use copies of -ods at each step of the construction. See Figures A-C for the standard universal dendrites , and .

The standard universal dendrites have the following properties.

1. For each is universal in the class of all dendrites for which the order of their ramification points is less than or equal to , see e.g. [Menger 1932, Chapter 10, § 6, p. 322].
2. If with , then there exists an open mapping of onto , [Chaaratonik 1980, Theorem 2, p. 492].
3. Among all standard universal dendrites only and are homeomorphic with all their open images, [Chaaratonik 1980, Corollary, p. 493].
4. For each a monotone surjection of onto itself is a near homeomorphism if and only if , [Charatonik 1991, Corollary 5.5, p. 178].
5. Any two standard universal dendrites and of some orders are monotonely equivalent, [Charatonik 1991, Corollary 6.6, p. 180].
6. For each the dendrite is monotonely homogeneous [Charatonik 1991, Theorem 7.1, p. 186].

Other mapping properties of the standard universal dendrites can be found e.g. in [Chaaratonik 1980], [Charatonik 1991], [Charatonik 1995], [Charatonik et al. 1997a], [Charatonik et al. 1998], [Charatonik et al. 1994] and [Charatonik W.J. et al. 1994].

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