{Preliminary properties}
\label{dendrites-preliminary}

The following properties of dendrites are known (see e.g. \cite[Chapter 10,
p. 165]{Nadler1992a},  \cite [\S 51, VI, p. 300]{Kuratowski1968a}
or \cite[p. 88]{Whyburn1942a}; compare
also \cite[Theorems 1.3 and 1.5, p. 233]{Charatonik+1998a}).
\rostere
\item Every subcontinuum of a dendrite is a dendrite (thus each dendrite is
hereditarily locally connected).
\item Every connected subset of a dendrite is arcwise connected.
\item In any dendrite $X$ the order of a point $p$ is equal to the number of
components of $X \setminus \{p\}$.
\item The set of all end points of a dendrite is $0$-dimensional.
\item The set of all end points of a dendrite is a $G_\delta$-set.
\item The set of all \gs{ordinary points}{ordinary point} (i.e., points of
order 2) of a dendrite $X$ is dense in $X$.
\item Every dendrite has at most countably many ramification points.
\item If $m$ is the order of a ramification point in a dendrite, then $m \in
\{3,4,\dots,\omega\}$.
\item No dendrite contains points of order $\aleph_0$ or $\mathfrak c$
(i.e., each dendrite is a \g{regular continuum}). \par

\item In any dendrite the set of all its end points is dense if and only if
the set of all its ramification points is dense.

\item Dendrites are \gs{absolute retracts}{absolute retract}. In fact, the
class of all 1-dimensional absolute retracts coincide with the class of
dendrites, \cite[Corollary 13.5, p. 138]{Borsuk1967a}.

\item A continuum $X$ is a dendrite if and only if for every compact space
(continuum) $Y$ and for every \gs{light}{light mapping}
\g{confluent mapping} $f: Y \to f(Y)$ such that $X \subset f(Y)$ there
is a copy $X'$ of $X$ in $Y$ for which the restriction $f|X': X' \to X$ is a
homeomorphism, \cite{Charatonik+XXXXb}.
\rosteree
