{Mapping hierarchy of dendrites}


A classification of topological spaces from the standpoint
of the theory of mappings (called also a \g{mapping hierarchy of spaces})
is defined in
\cite{Borsuk1959a}, and it is investigated for dendrites in
\cite{Charatonik+1994a}. Some questions asked there have
already been answered. We recall some examples related to
this subject.

\rostere
\item There exist two dendrites $X$ and $Y$ such that $\{X, Y\}$ has no
infimum and no supremum with respect to the class of
\gs{monotone mappings}{monotone mapping} (see \cite[Example
5.45, p. 18]{Charatonik+1994a}).
\item There exist dendrites $X$ and $Y$, both admitting
\gs{open mappings}{open mapping} onto arcs, such that
$Y \subset X$ and there is no open
mapping from $X$ onto $Y$ (see \cite[Example 6.67, p. 34]{Charatonik+1994a}) \par
Indeed, in an arm $A$ of a simple triod $Y$ fix a sequence $\{p_n\}$ of
points of $A$ converging to the center of the triod, and take a sequence
$\{A_n\}$ of arcs such that $Y \cap A_n = \{p_n\}$ and that $\lim
\diam A_n = 0$. Then $X = Y \cup \bigcup\{A_n: n \in \mathbb N\}$
is the needed dendrite.
\item There exists a dendrite which does not admit any
\g{open mapping} onto an arc and which is the union of two subdendrites
admitting open mappings onto an arc (see \cite[Example 6.69, p. 34]{Charatonik+1994a}).
%\item There exists an openly minimal dendrite that has two topologically
%different open images, see \cite[Proposition 3.5, p. 233]{Opela+2000a}. Openly
%minimal dendrites having a prescribed finite number of different open
%images are studied in \cite{Jelinek+2001a}.
\item There is a bounded chain of dendrites (ordered with respect to open
mappings) that has no supremum (see \cite{Podbrdsky+2001a}).
\item There is in the plane an uncountable family of dendrites every two
members of which are \gs{incomparable by $r$-mappings}{spaces
incomparable by a class of mappings} (see
\cite{Sieklucki1959a}).
\item There is a family of dendrites ordered with respect to $r$-mappings
similarly to the segment (see \cite{Sieklucki1961a}).
\rosteree
