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On self-homeomorphic spaces. (English) Zbl 0788.54040
A topological space \(X\) is called self-homeomorphic if for any open set \(U\) in \(X\) there is an open subset \(V\) of \(U\) such that \(V\) is homeomorphic to \(X\). Three stronger versions (in particular two pointwise ones) are considered, and interrelations between them are studied. After presenting some methods of constructing self-homeomorphic spaces, the authors study structures of the set of points at which a space is pointwise self-homeomorphic and of the set of local cut-points. Finally, self-homeomorphic dendrites are investigated. Universal dendrites are characterized and conditions are found under which an open mapping between the considered dendrites is possible. Several open problems are posed.

MSC:
54F15 Continua and generalizations
54C25 Embedding
54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites
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