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Recall that a space is said to be
incompressible if it is
homeomorphic to no proper subspace of itself. The circle
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 contains a subset such that
- A.
- each point of is fixed with respect to any homeomorphism of
onto a subcontinuum of ;
- B.
- each point of which is not an end point of lies in an arc
contained in and having its end points in ,
then is incompressible.
The
Miller dendrite
has the following properties, [Miller 1932].
- contains a subset having properties A and B above, so it is
incompressible.
- is not chaotic.
- 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: Dendrites of de Groot-Wille
Up: Dendrites
Previous: Chaotic and rigid dendrites
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih
2001-11-30