A Selection Theorem for Strongly Regular Multivalued Mappings |
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Authors: | Sergei M Ageev and Duan Repov |
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Institution: | (1) Department of Mathematics and Physics, Brest State Pedagogical Institute, 224665 Brest, Belorussia;(2) Institute for Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, P.O.B. 2964, 1001 Ljubljana, Slovenia |
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Abstract: | We prove the following generalization of a theorem of Ferry concerning selections of strongly regular multivalued maps onto the class of paracompact spaces: Let : X (Z, ) be a map of a paracompact space X into a metric space (Z, ) whose values (x) are complete subspaces of Z and absolute extensors (AE), for every x X. Suppose that can be represented as = , where : X Y is a continuous singlevalued map of X onto some finite-dimensional paracompact space Y and : Y (Z, ) is a strongly regular map. Then for every closed subset A X and every selection r : A Z of the map |A : A Z, there exists an extension
: X Z of r such that
is a selection of the map . We also prove a local version of this theorem. |
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Keywords: | strongly regular maps selection of multivalued maps absolute neighborhood extensors extensions of selections |
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