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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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