Geodesic
The JNR Property and the Borel Structure of a Banach Space
Serdica Mathematical Journal, Tome 26 (2000) no. 1, pp. 13-32

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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. Finally, by means of “good” injections into c0 (Γ), we give a great class of Banach spaces with the JNR property.
Keywords: Borel Sets, Countable Cover By Sets Of Small Local Diameter, Topological Invariants Of The Weak Topology 
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Oncina, L. The JNR Property and the Borel Structure of a Banach Space. Serdica Mathematical Journal, Tome 26 (2000) no. 1, pp. 13-32. http://geodesic.mathdoc.fr/item/SMJ2_2000_26_1_a2/