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On continuity and selections of multifunctions

Published online by Cambridge University Press:  17 April 2009

Pandelis Dodos
Pandelis Dodos
Affiliation:
Department of Mathematics, Faculty of Applied Sciences, National Technical University of Athens, Athens, Greece
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Abstract

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The notions of a Baire-1 and a weak Baire-1 multifunction are defined and a striking analogy between Baire-1 multifunctions and classical Baire-1 functions is established. A selection theorem is presented which asserts that if X is a metrisable space, Y a Polish space and F: X → 2Y/{∅} a closed-valued, weak Baire-1 multifunction, then F admits a Baire-1 selection. Using the machinery developed we prove that if X is a Banach space with separable dual, then every weak* usco, defined on a completely metrisable space Z, which values are weakly* compact subsets of the dual, is norm lower semicontinuous on a dense Gδ set.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 65 , Issue 3 , June 2002 , pp. 407 - 422
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Aliprantis, C.D. and Border, K.C., Infinite dimensional analysis, (Second edition) (Springer-Verlag, Berlin, 1999).CrossRefGoogle Scholar
[2]Deutsch, F., ‘A survey of continuous selections’, Contemp. Math. 18 (1983), 49–71.CrossRefGoogle Scholar
[3]Engelking, R., General topology, Sigma Series in Pure Mathematics 6, (Second edition) (Heldermann-Verlag, Berlin, 1989).Google Scholar
[4]Fort, M.K., ‘Category theorems’, Fund. Math. 42 (1955), 276–288.CrossRefGoogle Scholar
[5]Hu, S. and Papageorgiou, N.S., Handbook of multivalued analysis, Volume I: Theory, Mathematics and its Applications 419 (Kluwer, Dordrecht, 1997).Google Scholar
[6]Kechris, A.S., Classical descriptive set theory, Graduate Texts in Math. 156 (Springer-Verlag, New York, 1995).CrossRefGoogle Scholar
[7]Kenderov, P.S., ‘Semicontinuity of set valued monotone mappings’, Fund. Math. 88 (1975), 61–69.CrossRefGoogle Scholar
[8]Kuratowski, K., Topology, I (Academic Press, New York, London, 1966).Google Scholar
[9]Kuratowski, K. and Ryll-Nardzewski, C., ‘A general theorem on selectors’, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 397–403.Google Scholar
[10]Michael, E., ‘Continuous selections I’, Ann. Math. 63 (1956), 361–382.Google Scholar
[11]Odell, E. and Rosenthal, H.P., ‘A double dual characterization of separable Banach spaces containing l 1’, Israel J. Math. 20 (1975), 375–384.CrossRefGoogle Scholar