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Local homeomorphisms and related mappings on graphs. (English) Zbl 0597.54013

Let a class \({\mathcal S}\) of topological spaces and a class \({\mathcal M}\) of mappings be given. The following problem is posed: characterize all spaces \(X\in {\mathcal S}\) having the property that if \(f: X\to f(X)\) is a mapping belonging to \({\mathcal M}\) with non-degenerate f(X), then there exists a homeomorphism of X onto f(X). In other words, the question is to characterize all spaces in the class \({\mathcal S}\) such that their images under mappings from M are homeomorphic. We discuss the above problem for the class \({\mathcal S}\) of graphs taking as \({\mathcal M}\) the classes of open mappings, of local homeomorphisms and of local homeomorphisms in the large sense. Using these and some other mappings we get a characterization of an arc and several characterizations of a simple closed curve. Some unsolved problems are also stated.

MSC:

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

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