<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">{Other universal dendrites}
\label{ex-other-universal-dendrites}

For a given nonempty set $S \subset \{3, 4, \dots, \omega\}$
we denote by $D_S$
any dendrite $X$
satisfying the following two conditions:
\rosteri
\item[(a)] if $p$ is a ramification point of $X$, then $\ord(p,X)
\in S$;
\item[(b)] for each arc $A$ contained in $X$ and for every $m \in S$
there is in $A$ a point $p$ with $\ord(p,X) = m$.
\rosterii
It is shown in \cite[Theorem 6.2, p. 229]{CharatonikW+1994a}
that the
\es{dendrite $D_S$}{dendrite DS}
is topologically unique, i.e., if
two dendrites satisfy conditions (a) and (b) with the same
set $S \subset \{3, 4, \dots, \omega\}$, then they are
homeomorphic. The dendrite $D_S$ is called the
\es{standard universal dendrite of orders in $S$}{standard universal dendrite of orders in S}
. If $S$ is a singleton $\{m\}$,
then $D_S$ is just the
\gs{standard universal dendrite $D_m$}{standard universal dendrite Dm}
defined previously.

The following properties of dendrites $D_S$ are known (see \cite[Theorems
6.4 and 6.6-6.8, p. 230; Corollary 6.10, p. 232]{CharatonikW+1994a}).
\rostere
\item\label{ex-other-universal-dendrites-spsh}
For any nonempty subset $S \subset \{3, 4, \dots,
\omega\}$, the dendrite $D_S$ is \g{strongly pointwise self-homeomorphic}.
\item If $\omega \in S$, then the dendrite $D_S$ is
\gs{universal}{universal space}
for the family of all dendrites.
\item If the set $S$ is finite with $\max S = m$, then $D_S$ is
universal for the family of all dendrites having orders of ramification
points at most $m$.
\item\label{ex-other-universal-dendrites-univfinord}
If the set $S$ is infinite and $\omega \notin S$,
then $D_S$ is universal for the family of all dendrites having finite orders
of ramification points.
\item Nonconstant \gs{open images}{open mapping} of standard universal
dendrites $D_S$ are homeomorphic to $D_S$ if and only if $S$ is a nonempty
subset of $\{3, \omega\}$.
\item For any nonempty subset $S \subset \{3, 4, \dots, \omega\}$ and
for an arbitrary dendrite $Y$ there exists a \g{monotone mapping} from $D_S$ onto $Y$,
\cite[Theorem 2.22, p. 239]{Charatonik+1998a}.
\item\label{ex-other-universal-dendrites-DSmonohom}
For any nonempty subset $S \subset \{3, 4, \dots,
\omega\}$ the dendrite $D_S$ is \gs{monotonely homogeneous}{homogeneous
with respect to monotone mappings}, \cite[Theorem 3.3, p. 292]{Charatonik1996a}.
\rosteree
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