First class selectors for upper semi-continuous multifunctions.

*(English)*Zbl 0644.54014The 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.

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

##### MSC:

54C65 | Selections in general topology |

54C50 | Topology of special sets defined by functions |

54C05 | Continuous maps |

##### Keywords:

Borel selector; upper-semi-continuous multifunctions; closed-end fragmentation; first Borel class selectors for multifunctions; \(\sigma \)- discrete function; absolute extensor
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##### References:

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