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Baire class 1 selectors for upper semicontinuous set-valued maps. (English) Zbl 0822.54017

Summary: Let \(T\) be a metric space and \(X\) a Banach space. Let \(F: T\to X\) be a set-valued map assuming arbitrary values and satisfying the upper semicontinuity condition: \(\{t\in T\): \(F(t)\cap C\neq \emptyset\}\) is closed for each weakly closed set \(C\) in \(X\). Then there is a sequence of norm-continuous functions converging pointwise (in the norm) to a selection for \(F\). We prove a statement of similar precision and generality when \(X\) is a metric space.

MSC:

54C65 Selections in general topology
46B99 Normed linear spaces and Banach spaces; Banach lattices
54C60 Set-valued maps in general topology

Citations:

Zbl 0822.54018
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