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Wazewski universal dendrite

By the Wazewski universal dendrite D_\omega $ we mean a dendrite D_\omega $ such that each ramification point of D_\omega $ is of order \omega$ and for each arc A \subset D_\omega$ the set of ramification points of D_\omega $ which belong to A$ is dense in A$. Its construction, known from [Whyburn 1942, Chapter K, p. 137] (compare also [Menger 1932, Chapter X, Section 6, p. 318]), is the following.

Let X^{(\omega)}_1 = F_\omega$. At the midpoint c$ of each maximal free arc contained in X^{(\omega)}_1$ (obviously the arc is a straight line segment) attach a sufficiently small copy of F_\omega $ so that c$ is the only common point of X^{(\omega)}_1$ and of the attached copy. Denote by X^{(\omega)}_2$ the union of X^{(\omega)}_1$ and of all attached copies. Thus X^{(\omega)}_2$ is a dendrite. At the midpoint of each maximal free arc contained in X^{(\omega)}_2$ we perform the same construction, i.e., we attach a sufficiently small copy of F_\omega $ so that m$ is the only common point of X^{(\omega)}_2$ and of the attached copy. Denote by X^{(\omega)}_3$ the union of X^{(\omega)}_2$ and of all attached copies. Thus X^{(\omega)}_3$ is a dendrite. Continuing in this way we obtain an increasing sequence of dendrites X^{(\omega)}_1 \subset X^{(\omega)}_2 \subset \dots \subset X^{(\omega)}_n \subset X^{(\omega)}_{n+1} \subset \dots $. The construction can be done in the plane in such a way that the limit continuum D_\omega $ defined by

\displaystyle D_\omega = \mathrm{cl}\,(\bigcup \{X^{(\omega)}_n: n \in \mathbb{N}\})$

is again a dendrite. See Figure A.

Figure 1.3.6: ( A ) Wazewski universal dendrite D_\omega $
A.gif

For another construction of D_\omega $ (using inverse limits) see [Nadler 1992, 10.37, p. 181-185].

The following properties of D_\omega $ are known.

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

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

Here you can find source files of this example.

Here you can check the table of properties of individual continua.

Here you can read Notes or write to Notes ies of individual continua.
next up previous contents index
Next: Universal dendrites of order Up: Dendrites Previous: Universal dendrites
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30