The Point of Continuity Property: Descriptive Complexity and Ordinal Index

∗ Supported by D.G.I.C.Y.T. Project No. PB93-1142

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.