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

Zbl 0822.54018
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