next up previous contents index
Next: Dendrites of de Groot-Wille Up: Dendrites Previous: Chaotic and rigid dendrites

Miller dendrite

Recall that a space is said to be incompressible if it is homeomorphic to no proper subspace of itself. The circle S^1$ is an example of such a space. In 1925 Zarankiewicz [Zarankiewicz 1925] asked whether there exists an incompressible dendrite. The question has been answered in the positive in 1932 by E. W. Miller [Miller 1932]. The reader is referred to the original Miller's paper for the (rather long) construction of the example. The main idea of the proof is based on the following implication, [Miller 1932, Theorem, p. 831].

If a dendrite X$ contains a subset K$ such that

A.
each point of K$ is fixed with respect to any homeomorphism of X$ onto a subcontinuum of X$;
B.
each point of X$ which is not an end point of X$ lies in an arc contained in X$ and having its end points in K$,
then X$ is incompressible.

The Miller dendrite S$ has the following properties, [Miller 1932].

  1. S$ contains a subset K$ having properties A and B above, so it is incompressible.
  2. S$ is not chaotic.
  3. S$ contains open arcs as open subsets, hence it is not rigid.

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: Dendrites of de Groot-Wille Up: Dendrites Previous: Chaotic and rigid dendrites
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30