{Universal dendrites of order $m$}
\label{ex-universal-dendrite-m}
Let $m \in
\{3, 4, \dots, \omega\}$. By the
\es{standard universal dendrite of order $m$}{standard universal dendrite of order m}
 we mean a
\es{dendrite $D_m$}{dendrite Dm}
such
that each ramification point of $D_m$ is of order $m$ and
for each arc $A \subset D_m$ the set of ramification points
of $D_m$ which belong to $A$ is dense in $A$. Their
constructions, known again from \cite[Chapter K, p.
137]{Wazewski1923a} (see also \cite[(4), p.
168]{Charatonik1991a}; for the inverse limit construction
see \cite[p. 491]{Charatonik1980a}), mimic that of the
Wa\.zewski universal dendrite $D_\omega$, but instead of
copies of $F_\omega$ we use copies of $m$-ods at each step
of the construction.
%More precisely, we take $X^{(m)}_1$ as an $m$-od (being the union of
%straight line segments). Having defined $X^{(m)}_n$ for some $n \in
%\mathbb N$ (that is the union of straight line segments in the plane) we
%attach at the midpoint of each maximal free arc contained in $X^{(m)}_n$ a
%sufficiently small copy of $(m - 2)$-od so that $c$ is the only common
%point of $X^{(\omega)}_1$ and of the attached copy. Then $X^{(m)}_{n+1}$ is
%defined as the union of $X^{(m)}_n$ and of all the attached copies of $(m -
%2)$-ods. Note that each ramification point of $X^{(m)}_{n+1}$ is of order
%$m$ by construction. All this can be done in the plane in such a way that
%the limit continuum $D_\omega$ defined by
%$$D_m = \mathrm{cl}\,(\bigcup \{X^{(m)}_n: n \in \mathbb N\})$$
%is again a dendrite. In particular, if $m = \omega$, then the standard
%universal dendrite of order $\omega$ is just the Wa\.zewski dendrite
%$D_\omega$.
See Figures A--C for the standard universal dendrites $D_3$,
$D_4$ and $D_6$.

\FIGURE{A}{standard universal dendrite $D_3$}

\GIF{AA}{standard universal dendrite $D_3$ - an animation}

\FIGURE{B}{standard universal dendrite $D_4$}

\GIF{BB}{standard universal dendrite $D_4$ - an animation}

\GIF{BBB}{standard universal dendrite $D_4$ produced as an intersection - an animation}

\FIGURE{C}{standard universal dendrite $D_6$}


The standard universal dendrites $D_m$ have the following properties.
\rostere
\item For each $m \in \{3, 4, \dots, \omega\}$
$D_m$ is \gs{universal}{universal space} in the class of all
dendrites for which the order of their ramification points
is less than or equal to $m$, see e.g. \cite[Chapter 10, \S
6, p. 322]{Menger1932a}.
\item If $m, n \in \mathbb N$ with $3 \le m < n$, then there exists an
\g{open mapping} of $D_n$ onto $D_m$,
\cite[Theorem 2, p. 492]{Charatonik1980a}.
\item Among all standard universal dendrites $D_m$ only $D_3$ and
$D_\omega$ are homeomorphic with all their
\gs{open images}{open mapping}, \cite[Corollary, p.
493]{Charatonik1980a}.
\item For each $m \in \{3, 4, \dots, \omega\}$ a \gs{monotone surjection}
{monotone mapping} of $D_m$ onto itself is a \g{near
homeomorphism} if and only if $m = 3$, \cite[Corollary 5.5,
p. 178]{Charatonik1991a}.
\item\label{ex-ex-universal-dendrite-m-twomonoequi}
Any two standard universal dendrites $D_m$ and
$D_n$ of some orders $m, n \in \{3, 4, \dots, \omega\}$ are
\gs{monotonely equivalent}{monotonely equivalent spaces},
\cite[Corollary 6.6, p. 180]{Charatonik1991a}.
\item\label{ex-universal-dendrite-m-dmmonohom}
For each $m \in \{3, 4, \dots, \omega\}$ the
dendrite $D_m$ is \gs{monotonely homogeneous}{homogeneous
with respect to monotone mappings} \cite[Theorem 7.1, p.
186]{Charatonik1991a}.
\rosteree

Other mapping properties of the standard universal dendrites $D_m$ can be
found e.g. in \cite{Charatonik1980a}, \cite{Charatonik1991a},
\cite {Charatonik1995a}, \cite {Charatonik+1997a}, \cite{Charatonik+1998a},
\cite{Charatonik+1994a} and \cite{CharatonikW+1994a}.
