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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

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