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Borel selectors for upper semi-continuous multi-valued functions

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  26 February 2010

J. E. Jayne  and
C. A. Rogers
J. E. Jayne
Affiliation:
Department of Mathematics, University College London, London, WC1E 6BT
C. A. Rogers
Affiliation:
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT
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In our paper [12[ we made extensive use of the details of the proofs given in our earlier paper [11], and, in particular, we claimed that Lemma 3 of [11] holds, not just when Y is a metric space, but also when Y is a Hausdorff space, provided X × Y is a Fréchet space. In a corrigendum to [11], we give a corrected version of this Lemma 3, but it seems to depend, in an essential way, on the assumption that Y is a metric space, or at least a perfectly normal space. In this note we show that a modified version of this Lemma 3 enables us to justify all the theorems in [12] by use of a modified method of selection.

MSC classification

Secondary: 46B99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 32 , Issue 2 , December 1985 , pp. 324 - 337
Copyright
Copyright © University College London 1985

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References

1.Bessaga, C. and Pelczynski, A.. Selected Topics in Infinite-Dimensional Topology (PWN, Warsaw, 1975).Google Scholar
2.Borsuk, K.. Theory of Retracts, Mono. Mat. (PWN, Warsaw, 1967).Google Scholar
3.Borsuk, K.. Theory of Shape, Mono. Mat. (PWN, Warsaw, 1975).Google Scholar
4.Borsuk, K.. Uber Isomorphie der Funktional Raume. Bull Acad. Polon. Set., Ser. A, (1933), 1–10.Google Scholar
5.Dugundji, J.. An extension of Tietze's theorem. Pacific J. Math., 1 (1951), 353–367.Google Scholar
6.Edgar, G. A.. Measurability in a Banach space. Indiana Univ. Math. J., 26 (1977), 663–677.CrossRefGoogle Scholar
7.Hansell, R. W.. Borel measurable mappings for nonseparable metric spaces. Trans. Amer. Math. Soc, 161 (1971), 145–169.CrossRefGoogle Scholar
8.Hansell, R. W.. On Borel mappings and Baire functions. Trans. Amer. Math. Soc, 194 (1974), 195–211.Google Scholar
9.Hansell, R. W., Jayne, J. E., Labuda, I. and Rogers, C. A.. Boundaries of and selectors for upper semi-continuous set-valued functions. Math. Zeit., 189 (1985), 297–318.CrossRefGoogle Scholar
10.Hu, S. T.. Theory of Retracts (Detroit, 1965).Google Scholar
11.Jayne, J. E. and Rogers, C. A.. Upper semi-continuous set-valued functions. Acta Math., 149 (1982), 87–125, and a Corrigendum Acta Math. 155 (1985), 149–152.CrossRefGoogle Scholar
12.Jayne, J. E. and Rogers, C. A.. Borel selectors for upper semi-continuous multi-valued functions. J. Funct. Anal, 56 (1984), 279–299.CrossRefGoogle Scholar