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The JNR Property and the Borel Structure of a Banach Space

Oncina, L. — 2000

Serdica Mathematical Journal

Research partially supported by a grant of Caja de Ahorros del Mediterraneo. In this paper we study the property of having a countable cover by sets of small local diameter (SLD for short). We show that in the context of Banach spaces (JNR property) it implies that the Banach space is Cech-analytic. We also prove that to have the JNR property, to be σ- fragmentable and to have the same Borel sets for the weak and the norm topologies, they all are topological invariants of the weak topology....

Generalized first class selectors for upper semi-continuous set-valued maps in Banach spaces

R. W. Hansell;  L. Oncina — 2005

Czechoslovak Mathematical Journal

In this paper we deal with weakly upper semi-continuous set-valued maps, taking arbitrary non-empty values, from a non-metric domain to a Banach space. We obtain selectors having the point of continuity property relative to the norm topology for a large class of compact spaces as a domain. Exact conditions under which the selector is of the first Borel class are also investigated.

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