{Miller dendrite}

Recall that a space is said to be \gs{
incompressible}{incompressible space} if it is
homeomorphic to no proper subspace of itself. The circle
$S^1$ is an example of such a space. In 1925 Zarankiewicz
\cite{Zarankiewicz1925a} asked whether there exists an
incompressible dendrite. The question has been answered in
the positive in 1932 by E. W. Miller \cite{Miller1932a}. 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,
\cite[Theorem, p. 831]{Miller1932a}.

If a dendrite $X$ contains a subset $K$ such that
\rosteri
\item [A.] each point of $K$ is fixed with respect to any homeomorphism of
$X$ onto a subcontinuum of $X$;
\item [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 $K$,
\rosterii
then $X$ is incompressible.


The
\es{Miller dendrite $S$}{Miller dendrite S}
 has the following properties, \cite{Miller1932a}.
\rostere
\item $S$ contains a subset $K$ having properties A and B above, so it is
incompressible.
\item $S$ is not \gs{chaotic}{chaotic space}.
\item $S$ contains open arcs as open subsets, hence it is not \gs{rigid}{rigid space}.
\rosteree
