First class selectors for upper semi-continuous multifunctions. (English) Zbl 0644.54014

The paper improves selection theorems for arbitrary upper-semi-continuous multifunctions due to J. Jayne and C. A. Rogers. A concept of closed-end fragmentation of a Hausdorff space helps to find a sufficiently strong result to prove the existence of first Borel class selectors for multifunctions from a metric space X to (1) a metric space Y, or (2) the Banach space Y with its weak topology. Some strengthening for separable spaces and for \(c_ 0(\Gamma)\) is made.
The selector is chosen to be a \(\sigma\)-discrete function from X to Y, and it is shown that some former results easily imply that such functions are in the first Baire class if Y is “absolute extensor for metric spaces”. It is pointed out that the author’s former proof of the stronger result, with Y being the extensor for X only, was not correct. The validity of this stronger claim remained unanswered.
Reviewer: P.Holicky


54C65 Selections in general topology
54C50 Topology of special sets defined by functions
54C05 Continuous maps
Full Text: DOI


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