On the Cantor-Bendixson derivative, resolvable ranks, and perfect set theorems of A. H. Stone

A Borel derivative on the hyperspace 2(X) of a compactum X is a Borel monot one map D : 2(X) --> 2(X). The derivative determines a Cantor-Bendixson typ e rank delta : 2(X) --> omega(1) boolean OR {infinity}. We show that if A s ubset of 2(X) is analytic and Z subset of A intersects stationary many laye rs delta(-1)({xi}), then for almost all xi, A boolean AND delta(-1)({xi}) c annot be separated from Z boolean AND boolean OR alpha<xi (delta-1)({alpha} ) land also from Z boolean AND boolean ORalpha>xi (delta-1)({alpha})) by an y F-sigma-set. Another main result involves a natural partial order on 2(X) related to the derivative. The results are obtained in a general framework of "resolvable ranks" introduced in the paper.