Compact Covering Mappings between Borel Sets and the Size of Constructible Reals

We prove that the topological statement: “Any compact covering mapping between two Borel sets is inductively perfect” is equivalent to the set-theoretical statement: “ ∀α ∈ ωω , א 1 < א1”. The starting point of this work is the following topological problem: Problem. Is any compact covering mapping between two Borel spaces inductively perfect? We recall that if f : X → Y is a continuous and onto mapping, then: – f is said to be perfect if the inverse image by f of any compact subset of Y is compact. – f is said to be compact covering if any compact subset of Y is the direct image of some compact subset of X . – f is said to be inductively perfect if there exists a subset X ′ of X such that the restriction of f to X ′ is a perfect mapping from X ′ onto Y . All spaces considered here will implicitly be supposed to be embedded in 2. We follow standard logical notation: Σ1, Π 1 1, ∆ 1 1, for the classes of analytic, coanalytic, Borel sets; Σξ , Π 0 ξ for the additive and multiplicative Borel classes; Σ 1 1 , Π 1 1, ∆ 1 1, Σ ξ , Π 0 ξ for the effective versions of these classes (see [7] for more details). Notice that Π2 = Gδ and Σ 0 2 = Kσ (i.e. σ-compact, since our spaces are embedded in 2). It is clear that any inductively perfect mapping is compact covering. We present below the main known results for the converse implication: (A) In ZFC: – Any compact covering mapping from a Π2 space onto any space is inductively perfect. – Any compact covering mapping from any space onto a Σ2 space is inductively perfect. (B) If we assume Det (Σ1) (Σ 1 1 determinacy), then: Any compact covering mapping from a Π1 space onto a Π 1 1 space is inductively perfect. Received by the editors May 31, 2001. 2000 Mathematics Subject Classification. Primary 03E15; Secondary 03E45, 54H05. c ©2003 American Mathematical Society