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Universal dendrites of order m$

Let m \in \{3,4,\dots,\omega\}$. By the standard universal dendrite of order m$ we mean a dendrite D_m$ such that each ramification point of D_m$ is of order m$ and for each arc A \subset D_m$ the set of ramification points of D_m$ which belong to A$ is dense in A$. 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 D_\omega $, but instead of copies of F_\omega $ we use copies of m$-ods at each step of the construction. See Figures A-C for the standard universal dendrites D_3$, D_4$ and D_6$.

Figure 1.3.7: ( A ) standard universal dendrite D_3$
A.gif

Figure: ( AA ) standard universal dendrite D_3$ - an animation
AA.gif

Figure 1.3.7: ( B ) standard universal dendrite D_4$
B.gif

Figure: ( BB ) standard universal dendrite D_4$ - an animation
BB.gif

Figure: ( BBB ) standard universal dendrite D_4$ produced as an intersection - an animation
BBB.gif

Figure 1.3.7: ( C ) standard universal dendrite D_6$
C.gif

The standard universal dendrites D_m$ have the following properties.

  1. For each m \in \{3,4,\dots,\omega\}$ D_m$ is universal in the class of all dendrites for which the order of their ramification points is less than or equal to m$, see e.g. [Menger 1932, Chapter 10, § 6, p. 322].
  2. If m, n \in \mathbb{N}$ with 3 \le m < n$, then there exists an open mapping of D_n$ onto D_m$, [Chaaratonik 1980, Theorem 2, p. 492].
  3. Among all standard universal dendrites D_m$ only D_3$ and D_\omega $ are homeomorphic with all their open images, [Chaaratonik 1980, Corollary, p. 493].
  4. For each m \in \{3,4,\dots,\omega\}$ a monotone surjection of D_m$ onto itself is a near homeomorphism if and only if m = 3$, [Charatonik 1991, Corollary 5.5, p. 178].
  5. Any two standard universal dendrites D_m$ and D_n$ of some orders m, n \in \{3, 4, \dots, \omega\}$ are monotonely equivalent, [Charatonik 1991, Corollary 6.6, p. 180].
  6. For each m \in \{3,4,\dots,\omega\}$ the dendrite D_m$ is monotonely homogeneous [Charatonik 1991, Theorem 7.1, p. 186].

Other mapping properties of the standard universal dendrites D_m$ 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].

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