×

zbMATH — the first resource for mathematics

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

MSC:
54C65 Selections in general topology
54C50 Topology of special sets defined by functions
54C05 Continuous maps
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arens, R; Eells, J, On embedding uniform and topological spaces, Pacific J. math., 6, 397-403, (1956) · Zbl 0073.39601
[2] {\scJ. Ceder and S. Levi}, On the search for Borel 1 selections, Časopis Pěst. Mat., in press. · Zbl 0575.28008
[3] Choquet, G, Convergences, Ann. univ. Grenoble, 23, 57-112, (1947-1948)
[4] Dolecki, S; Lechicki, A, On structure of upper semicontinuity, J. math. anal. appl., 88, 547-554, (1982) · Zbl 0503.54023
[5] Dugundji, J, An extension of Tietze’s theorem, Pacific J. math., 1, 353-367, (1951) · Zbl 0043.38105
[6] Edgar, G.A; Wheeler, R.F, Topological properties of Banach spaces, Pacific J. math., 115, 317-350, (1984) · Zbl 0506.46007
[7] Hansell, R.W, A measurable selection and representation theorem in non-separable spaces, () · Zbl 0313.54044
[8] Hansell, R.W, Borel measurable mappings for nonseparable metric spaces, Trans. amer. math. soc., 161, 145-169, (1971) · Zbl 0232.28007
[9] Hansell, R.W, On Borel mappings and Baire functions, Trans. amer. math. soc., 194, 195-211, (1974) · Zbl 0295.54047
[10] Hansell, R.W; Jayne, J.E; Labuda, I; Rogers, C.A, Boundaries of and selectors for upper semi-continuous multi-valued maps, Math. Z., 189, 297-318, (1985) · Zbl 0544.54016
[11] Hansell, R.W; Jayne, J.E; Talagrand, M, First class selectors for weakly upper semi-continuous multivalued maps in Banach spaces, J. reine angew. math., 361, 201-220, (1985) · Zbl 0573.54012
[12] Jayne, J.E; Rogers, C.A, Upper semi-continuous set-valued functions, Acta math., 149, 87-125, (1982) · Zbl 0523.54013
[13] Jayne, J.E; Rogers, C.A, Borel selectors for upper semi-continuous multi-valued functions, J. funct. anal., 56, 279-299, (1984) · Zbl 0581.28007
[14] Jayne, J.E; Rogers, C.A, Borel selectors for upper semi-continuous set-valued maps, Acta math., 155, 41-79, (1985) · Zbl 0588.54020
[15] {\scJ. E. Jayne and C. A. Rogers}, Borel selectors for upper semi-continuous multi-valued functions: Corrigendum, preprint. · Zbl 0581.28007
[16] Kuratowski, K, ()
[17] Maitra, A; Rao, B.V, Selection theorems and the reduction principle, Trans. amer. math. soc., 202, 57-66, (1975) · Zbl 0314.54019
[18] {\scV. V. Srivatsa}, Baire class 1 selectors for upper semi-continuous set-valued maps, to appear. · Zbl 0822.54017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.