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Modifications of the Gehman dendrite

The concept of the Gehman dendrite has been generalized in [Arévalo et al. 2001, Section 4, p. 4-10] as follows.

  1. For each natural number n \ge 3$ there exists topologically unique dendrite G_n$ dendrite G_n$ such that
    1. \mathrm{ord}\, (p, G_n) = n$ for each point p \in R(G_n)$;
    2. E(G_n)$ is homeomorphic to the Cantor ternary set.

    (Note that G_3$ is just the Gehman dendrite G$.)

  2. There exists a dendrite G_{\omega}$ such that
    1. \mathrm{ord}\,(p, G_{\omega})$ is finite for each point p \in G_{\omega}$;
    2. E(G_{\omega})$ is homeomorphic to the Cantor ternary set;
    3. for each natural number n$ and for each maximal arc A$ contained in G_{\omega}$ there is a point q \in A$ such that \mathrm{ord}\,(q, G_{\omega}) \ge n$.

    See Figure A.

    Figure 1.3.19: ( A ) G_4$
    A.gif

    The dendrites G_n$ and G_{\omega}$ have the following universality properties.

  3. For each natural number n \ge 3$ the dendrite G_n$ is universal for the class of all dendrites X$ such that \mathrm{cl}\,E(X) = E(X)$ and that \mathrm{ord}\,(p,X) \le n$ for each point p \in X$.
  4. Each dendrite G_{\omega}$ is universal for the class of all dendrites with the closed set of end points.

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: Mapping hierarchy of dendrites Up: Dendrites Previous: Gehman dendrite
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30