Hurewicz-like Tests for Borel Subsets of the Plane

Let ξ ≥ 1 be a countable ordinal. We study the Borel subsets of the plane that can be made Πξ by refining the Polish topology on the real line. These sets are called potentially Πξ . We give a Hurewicz-like test to recognize potentially Πξ sets. 1. Preliminaries in dimension one Let us recall some results in dimension one before studying Borel subsets of the plane. In descriptive set theory, a standard way to see that a set is complicated is to note that it is more complicated than a well-known example. For instance, we have the following result (see [SR]): Theorem 1 (Hurewicz). Let Pf := {α ∈ 2/∃n ∈N ∀m≥ n α(m) = 0}, X be a Polish space, and A a Borel subset of X. Then exactly one of the following holds: (a) The set A is Π2(X). (b) There is u : 2→X continuous and one-to-one with Pf =u−1(A). This result has been generalized to the other Baire classes (see [Lo-SR]). We state this generalization in two parts: Theorem 2 (Louveau-Saint Raymond). Let ξ < א1, A1+ξ ∈ Σ1+ξ(2), X be a Polish space, and A, B disjoint analytic subsets of X. One of the following holds: (a) The set A is separable from B by a Π1+ξ(X) set. (b) There is u : 2→X continuous with A1+ξ⊆u(A) and 2\A1+ξ⊆u(B). If we moreover assume that A1+ξ / ∈Π1+ξ, then this is a dichotomy (in this case, and if ξ≥2, we can have u one-to-one). Theorem 3. There is a concrete example of A1+ξ ∈ Σ1+ξ(2) \ Π1+ξ(2), for ξ<א1. If we replace Pf (resp., Π2) with the set A1+ξ given by Theorem 3 (resp., Π 0 1+ξ), we get the generalization of Theorem 1 for ξ ≥ 2. We state this generalization in two parts for the following reasons: Received by the editors July 29, 2005. 2000 Mathematics Subject Classification. Primary 03E15; Secondary 54H05.