Wadge Hierarchy of differences of Co-analytic Sets

We begin the fine analysis of non Borel pointclasses. Working in ZFC + ̃ 11), we describe the Wadge hierarchy of the class of increasing differences of coanalytic subsets of the Baire space by extending results obtained by Louveau ([5]) for the Borel sets. Introduction. Collections of substets of the Baire space, the "logician’s reals", that are closed under continuous preimages have always been ubiquitous in descriptive set theory. It is thus quite remarkable to realize that the concept of pointclass has not been singled out and studied for itself before the 1960’s and the work of Wadge. In his PhD Thesis ([10]), he was the first to study systematically the concept, via the notion of continuous reducibility. Given two subsets A and B of the Baire space, A is said to be reducible to B, and we write A ≤W B, if and only if A is the preimage of B for some continuous function f from the Baire space to itself. The relation ≤W is merely by definition a preorder, and its initial segments are exactly the pointclasses of the Baire space. When restricted to a class with suitable closure properties, the preorder induced by ≤W on its equivalence classes, the Wadge degrees, is in fact a well-quasi-ordering. The study of this well-quasi-ordering, the Wadge hierarchy, and of the Wadge degreees gives thus the finest analysis of the pointclasses of the Baire space. The Wadge hierarchy of the Borel subsets of the Baire space has been thoroughly studied by Louveau in [5] and Duparc in [3] and [2], in two different manners that were both initiated by Wadge. The former relies on a Theorem proved by Wadge stating that all the non self-dual Borel pointclasses can be obtained by ω-ary Borel boolean operations on open sets a result later generalized to all non self-dual pointclasses of the Baire space by Van Wesep in [8] under AD, using of course arbitrary ω-ary boolean operations. Louveau’s work provides a description of all the Borel pointclasses, and thus of the whole Wadge hierarchy on the Borel sets, by means of boolean operations. The latter approach, followed by Duparc and that we do not pursue here, aims to define and use specific operations on sets in order to give for each Wadge class of Borel subsets a canonical complete set. This research was partially supported by the Swiss National Science Fundation grant number 200021− 135401. c © 0000, Association for Symbolic Logic 0022-4812/00/0000-0000/$00.00