On the Cantor-Bendixon derivative and perfect set theorems of A. H. Stone
by
R. Pol
Warsaw University and Miami University

Let 2^X be the space of compact subsets of a compact metric space X with the Hausdorff distance. A Borel derivative D : 2^X --> 2^X is a Borel map with DK subset K. Let D^\alpha : 2^X --> 2^X be the \alphath iterate of D, \alpha being an ordinal. The derivative D determines a tree-order on 2^X, K \prec L iff K = D^\alpha L for some \alpha >= 1, and the index map \delta: 2^X --> \omega₁ \cup {\infty}, where \delta(K) is the minimal \alpha with D^{(\alpha+ 1)}K = \emptyset or \infty, if no such \alpha exists. An important example is the Cantor-Bendixon derivative which assigns to each K the set K' of its accumulation points.

The following result was recently obtained jointly with J. Chaber.

Theorem Let D : 2^X --> 2^X be a Borel derivative and let \prec, \delta be the tree-order and the index determined by D. For each Souslin set S subset 2^X with \delta(S) stationary in \omega₁ there exists an \prec-antichain A subset S such that A \cap \delta^-1(\alpha) is a Cantor set for all but non-stationary many \alpha.

The Souslin sets in 2^X are continuous images of the irrationals.

This can be put in a more general framework, considering triples (E, \prec, \delta), where \prec is a tree-order on a separable metrizable space, \delta: E --> \omega₁ is \prec-monotone and the triple has some descriptive properties expressed in terms of functions \phi: E --> B(\omega₁) in the Baire space of weight \aleph₁.

The results are based on some links between certain classical topics in the descriptive set theory and the fine stucture of Borel sets in B(\omega₁), first investigated by A.H. Stone. This approach was considered by Gruenhage, Chaber and myself in joint papers ``On a perfect set theorem of A.H. Stone and N.N. Lusin's constituents'', Fund. Math. 148 (1995) and ``On Souslin sets and embedding in integer-valued functions on \omega₁'', Top. Appl. (to appear).

Date received: February 19, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caam-11.