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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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