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Selections and representations of multifunctions in paracompact spaces

Alberto BressanGiovanni Colombo — 1992

Studia Mathematica

Let (X,T) be a paracompact space, Y a complete metric space, F : X 2 Y a lower semicontinuous multifunction with nonempty closed values. We prove that if T + is a (stronger than T) topology on X satisfying a compatibility property, then F admits a T + -continuous selection. If Y is separable, then there exists a sequence ( f n ) of T + -continuous selections such that F ( x ) = f n ( x ) ; n 1 ¯ for all x ∈ X. Given a Banach space E, the above result is then used to construct directionally continuous selections on arbitrary subsets of ℝ × E.

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