Classifying Countable Boolean Terms

Let {Σα}α<ω1 , where ω1 is the first non-countable ordinal, denote the Borel hierarchy of subsets of the Cantor space 2. As usual, Πα denotes the dual class for Σα while ∆ 0 α = Σ 0 α ∩Πα — the corresponding ambiguous class. Let B = ∪α<ω1Σα denote the class of all Borel sets. The levels of Borel hierarchy, as well as many other classes of interest for descriptive set theory, may be defined by means of suitable (in general, infinitary) set-theoretic operations. Let us define a natural class of such operations, which we call ω1-terms, by induction: constants 0, 1 and variables vk(k < ω) are ω1-terms; if ti(i < ω) are ω1-terms, then so are the expressions t̄0, t0 ∪ t1, t0 ∩ t1, ∪i<ωti and ∩i<ωti. If t = t(vk) is an ω1-term, let t({Ak}) denote the value of t when each variable vk(k < ω) is interpreted as some set Ak ⊆ 2. Let t(Σ1) be the set of all values t({Ak}), when Ak ∈ Σ1 for any k < ω. We use similar notation t(C) also for other kinds of terms t and classes of sets C. It turns out that the classes t(Σ1) have a very natural description in terms of the m Wadge reducibility on the class P (2) of all subsets of 2 defined as follows: A ≤W B ↔ A = f−1(B), for some continuous function f : 2 → 2. By a Wadge class we mean any principal ideal of the form {X|X ≤W A}, for a given A ⊆ 2. Such a class is called Borel if A is Borel, and is called non-selfdual if A 6≤W Ā, where A = 2\A is the complement of A. The next fact was proved in [2, 3] (the last equivalence was observed in [1]).