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Continuous selections of finite-set valued mappings. (English) Zbl 0769.54021
This paper deals with the existence of continuous selections for finite- valued continuous multifunctions. Main theorem: Let \(X\) be a locally connected treelike space and \(Y\) a Hausdorff topological space. Then each continuous finite-valued mapping \(F: X\to 2^ Y\) admits continuous selections in the strong sense (i.e. you can choose a point in the graph). A counterexample shows that hereditary unicoherence of \(X\) is necessary. It is shown that “treelike” and “hereditarily unicoherent” are equivalent for various subclasses of connected and locally connected spaces.
Reviewer: C.R.Borges (Davis)
MSC:
54C65 Selections in general topology
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