F U N D a M E N T a Mathematicae 159 (1999)

We study properties of Σ1 and Π 1 1 subsets of ω ω that are cofinal relative to the orders ≤ (≤?) of full (eventual) domination. We apply these results to prove that the topological statement “Any compact covering mapping from a Borel space onto a Polish space is inductively perfect” is equivalent to the statement “∀α ∈ ω , ω ∩ L(α) is bounded for ≤?”. This work is a continuation of [3], in which we studied the validity of the following statement for two separable metric spaces X and Y : A(X,Y ): “Any compact covering mapping f : X → Y is inductively perfect”. We recall that if f : X → Y is a continuous mapping then f is said to be: • compact covering if any compact subset of Y is the direct image of some compact subset of X; • perfect if the inverse image of any compact subset of Y is a compact subset of X; • inductively perfect if there exists a subset X ′ of X such that the restriction of f to X ′ is a perfect mapping onto Y . Notice that, as we showed in [3], the study of these notions can easily be reduced to the case where all the spaces are zero-dimensional, hence subsets of ω or 2. Obviously, any inductively perfect mapping is compact covering: If K is any compact subset of Y then the set H = X ′ ∩ f−1(K) is compact and clearly f(H) = K. The converse statement, that is, A(X,Y ), is false in general but holds under some regularity assumptions on X or Y . The main known results in this direction are the following: 1991 Mathematics Subject Classification: Primary 03E15; Secondary 03E45, 54H05.