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Other universal dendrites

For a given nonempty set S \subset \{3, 4, \dots, \omega\}$ we denote by D_S$ any dendrite X$ satisfying the following two conditions:

(a)
if p$ is a ramification point of X$, then \mathrm{ord}\,(p,X)
\in S$;
(b)
for each arc A$ contained in X$ and for every m \in S$ there is in A$ a point p$ with \mathrm{ord}\,(p,X) = m$.
It is shown in [Charatonik W.J. et al. 1994, Theorem 6.2, p. 229] that the dendrite D_S$ is topologically unique, i.e., if two dendrites satisfy conditions (a) and (b) with the same set S \subset \{3, 4, \dots, \omega\}$, then they are homeomorphic. The dendrite D_S$ is called the standard universal dendrite of orders in S$ . If S$ is a singleton \{m\}$, then D_S$ is just the standard universal dendrite D_m$ defined previously.

The following properties of dendrites D_S$ 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 S \subset \{3, 4, \dots, \omega\}$, the dendrite D_S$ is strongly pointwise self-homeomorphic.
  2. If \omega \in S$, then the dendrite D_S$ is universal for the family of all dendrites.
  3. If the set S$ is finite with \max S = m$, then D_S$ is universal for the family of all dendrites having orders of ramification points at most m$.
  4. If the set S$ is infinite and \omega \notin S$, then D_S$ is universal for the family of all dendrites having finite orders of ramification points.
  5. Nonconstant open images of standard universal dendrites D_S$ are homeomorphic to D_S$ if and only if S$ is a nonempty subset of \{3, \omega\}$.
  6. For any nonempty subset S \subset \{3, 4, \dots, \omega\}$ and for an arbitrary dendrite Y$ there exists a monotone mapping from D_S$ onto Y$, [Charatonik et al. 1998, Theorem 2.22, p. 239].
  7. For any nonempty subset S \subset \{3, 4, \dots, \omega\}$ the dendrite D_S$ is monotonely homogeneous, [Charatonik 1996, Theorem 3.3, p. 292].

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