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 ,

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.