A Cantor-Bendixson-like process which detects Delta_2^0

TL;DR

This paper introduces a Cantor-Bendixson-like process to characterize the Delta_2^0 class in Baire space, providing new insights and alternative proofs related to descriptive set theory.

Contribution

It develops a novel process similar to Cantor-Bendixson to characterize Delta_2^0 sets and offers alternative proofs for known results in descriptive set theory.

Findings

New characterization of Delta_2^0 sets in Baire space

Alternative proofs for Wadge's results on guessability

A process analogous to Cantor-Bendixson for descriptive set classes

Abstract

For each subset of Baire space, we define, in away similar to a common proof of the Cantor-Bendixson Theorem, a sequence of decreasing subsets S_alpha of N^N, indexed by ordinals. We use this to obtain two new characterizations of the boldface Delta_2^0 Borel pointclass. ADDENDUM: In January 2012 we learned that the notion of guessability appeared in an equivalent form, and even with the same name, in the doctoral dissertation of William Wadge [4]. As for the main result of this paper, Wadge proved one direction and gave a proof for the other direction which he attributed to Hausdorff. The proofs in this paper present an alternate means to those results.