Geodesic
The Point of Continuity Property: Descriptive Complexity and Ordinal Index
Serdica Mathematical Journal, Tome 24 (1998) no. 2, pp. 199-214.

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Let X be a separable Banach space without the Point of Continuity Property. When the set of closed subsets of its closed unit ball is equipped with the standard Effros-Borel structure, the set of those which have the Point of Continuity Property is non-Borel. We also prove that, for any separable Banach space X, the oscillation rank of the identity on X (an ordinal index which quantifies the Point of Continuity Property) is determined by the subspaces of X with a finite-dimensional decomposition. If X does not contain l1 , subspaces with basis suffice. If X ∗ is separable, one can even restrict to subspaces with shrinking basis.
Keywords: Point of Continuity Property, Borel Set, Ordinal Index 
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Bossard, Benoit; López, Ginés. The Point of Continuity Property: Descriptive Complexity and Ordinal Index. Serdica Mathematical Journal, Tome 24 (1998) no. 2, pp. 199-214. http://geodesic.mathdoc.fr/item/SMJ2_1998_24_2_a6/