Mapping hierarchy of dendrites

A classification of topological spaces from the standpoint of the theory of mappings (called also a mapping hierarchy of spaces) is defined in [Borsuk 1959], and it is investigated for dendrites in [Charatonik et al. 1994]. Some questions asked there have already been answered. We recall some examples related to this subject.

1. There exist two dendrites and such that has no infimum and no supremum with respect to the class of monotone mappings (see [Charatonik et al. 1994, Example 5.45, p. 18]).
2. There exist dendrites and , both admitting open mappings onto arcs, such that and there is no open mapping from onto (see [Charatonik et al. 1994, Example 6.67, p. 34])

   Indeed, in an arm of a simple triod fix a sequence of points of converging to the center of the triod, and take a sequence of arcs such that and that . Then is the needed dendrite.

3. There exists a dendrite which does not admit any open mapping onto an arc and which is the union of two subdendrites admitting open mappings onto an arc (see [Charatonik et al. 1994, Example 6.69, p. 34]).
4. There is a bounded chain of dendrites (ordered with respect to open mappings) that has no supremum (see [Podbrdský et al. 2001]).
5. There is in the plane an uncountable family of dendrites every two members of which are incomparable by -mappings (see [Sieklucki 1959]).
6. There is a family of dendrites ordered with respect to -mappings similarly to the segment (see [Sieklucki 1961]).

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