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## Wazewski universal dendrite

By the Wazewski universal dendrite 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 . Its construction, known from [Whyburn 1942, Chapter K, p. 137] (compare also [Menger 1932, Chapter X, Section 6, p. 318]), is the following.

Let . At the midpoint of each maximal free arc contained in (obviously the arc is a straight line segment) attach a sufficiently small copy of so that is the only common point of and of the attached copy. Denote by the union of and of all attached copies. Thus is a dendrite. At the midpoint of each maximal free arc contained in we perform the same construction, i.e., we attach a sufficiently small copy of so that is the only common point of and of the attached copy. Denote by the union of and of all attached copies. Thus is a dendrite. Continuing in this way we obtain an increasing sequence of dendrites . The construction can be done in the plane in such a way that the limit continuum defined by is again a dendrite. See Figure A. For another construction of (using inverse limits) see [Nadler 1992, 10.37, p. 181-185].

The following properties of are known.

1. is universal in the class of all dendrites (see e.g. [Nadler 1992, 10.37, p. 181-185]).
2. is embeddable in the plane (in fact, it is constructed in the plane).
3. Each open image of is homeomorphic to (see [Chaaratonik 1980, Theorem 1, p. 490]).
4. is homogeneous with respect to monotone mappings, [Charatonik 1991, Theorem 7.1, p. 186].

For other mapping properties of , in particular ones related to the action of the group of autohomeomorphisms on , see [Charatonik 1995].

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Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30