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Monotone mappings of universal dendrites. (English) Zbl 0726.54012
The author first gives some auxiliary properties of local dendrites and then recalls, for $$m\in \{3,4,...,\omega \}$$, a construction of $$D_ m$$, the standard universal dendrite of order m, first given in T. Ważewski’s doctoral dissertation and later refined and simplified by K. Menger. After showing some of the basic properties $$D_ m$$, the author studies mappings of $$D_ m$$ onto itself that are either homeomorphisms, near homeomorphisms or monotone mappings. Starting with some results of H. Kato for $$D_ 3$$, the author generalizes these results to $$D_ m$$. For example, Kato showed that for each two points of order 3 of $$D_ 3$$, there exists a homeomorphism of $$D_ 3$$ onto itself mapping one of the points onto the other if and only if the two points are of the same Menger-Urysohn order. The author extends this result to all universal dendrites of order $$m\in \{3,4,...,\omega \}$$. Kato also showed that every monotone mapping of $$D_ 3$$ onto itself is a near homeomorphism and that $$D_ 3$$ is homogeneous with respect to monotone mappings. The author strengthens these results as follows. Theorem 1. For each $$m\in \{4,5,...,\omega \}$$, there exists a monotone mapping of $$D_ m$$ onto itself which is not a near homeomorphism. - Theorem 2. Every standard universal dendrite $$D_ m$$ of order $$m\in \{3,4,...,\omega \}$$ is homogeneous with respect to monotone mappings. - The author also studies dendrites which are monotone equivalent to the standard universal dendrites of order $$m\in \{3,4,...,\omega \}$$.

##### MSC:
 54C10 Special maps on topological spaces (open, closed, perfect, etc.) 54F50 Topological spaces of dimension $$\leq 1$$; curves, dendrites
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