{Wa\.zewski universal dendrite}
By the
\es{Wa\.zewski universal dendrite}{Wazewski universal dendrite}
$D_\omega$ we mean a dendrite $D_\omega$ such that
each ramification point of $D_\omega$ is of order $\omega$
and for each arc $A
\subset D_\omega$ the set of ramification points of
$D_\omega$ which belong to $A$ is dense in $A$. Its
construction, known from \cite[Chapter K, p.
137]{Whyburn1942a} (compare also \cite[Chapter X, Section 6,
p. 318]{Menger1932a}), is the following.

Let $X^{(\omega)}_1 = F_\omega$. At the midpoint $c$ of each
maximal \g{free arc} contained in $X^{(\omega)}_1$
(obviously the arc is a straight line segment) attach a
sufficiently small copy of $F_\omega$ so that $c$ is the
only common point of $X^{(\omega)}_1$ and of the attached
copy. Denote by $X^{(\omega)}_2$ the union of
$X^{(\omega)}_1$ and of all attached copies. Thus
$X^{(\omega)}_2$ is a dendrite. At the midpoint of each
maximal free arc contained in $X^{(\omega)}_2$ we perform
the same construction, i.e., we attach a sufficiently small
copy of $F_\omega$ so that $m$ is the only common point of
$X^{(\omega)}_2$ and of the attached copy. Denote by
$X^{(\omega)}_3$ the union of $X^{(\omega)}_2$ and of all
attached copies. Thus $X^{(\omega)}_3$ is a dendrite.
Continuing in this way we obtain an increasing sequence of
dendrites $X^{(\omega)}_1 \subset X^{(\omega)}_2 \subset
\dots \subset X^{(\omega)}_n \subset X^{(\omega)}_{n+1}
\subset \dots $. The construction can be done in the plane
in such a way that the limit continuum $D_\omega$ defined by
$$D_\omega = \cl(\bigcup \{X^{(\omega)}_n: n \in \mathbb N\})$$
is again a dendrite. See Figure A.

\FIGURE{A}{Wa\.zewski universal dendrite $D_\omega$}

%Figure 13.3.1 here


For another construction of $D_\omega$ (using inverse limits) see
\cite[10.37, p. 181-185]{Nadler1992a}.

The following properties of $D_\omega$ are known.
\rostere
\item $D_\omega$ is \gs{universal}{universal space} in the class of all
dendrites (see e.g. \cite[10.37, p. 181-185]{Nadler1992a}).
\item $D_\omega$ is embeddable in the plane (in fact, it is constructed in
the plane).
\item Each \gs{open image}{open mapping} of $D_\omega$ is homeomorphic to
$D_\omega$ (see \cite[Theorem 1, p. 490]{Charatonik1980a}).
\item $D_\omega$ is \gs{homogeneous with respect to monotone mappings}
{homogeneous with respect to a class of mappings},
\cite[Theorem 7.1, p. 186]{Charatonik1991a}.
\rosteree

For other mapping properties of $D_\omega$, in particular ones related to
the action of the group of autohomeomorphisms on $D_\omega$,
see \cite{Charatonik1995a}.
