Point of Continuity Property Borel Set Ordinal Index
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Mathematical Journal, Vol. 24, No 2, (1998), 199p-214p
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.