Hurewicz tests: separating and reducing analytic sets in a concise way

If we add the well-known fact that any subset N of the Cantor set C homeomorphic to the irrationals is never Fσ, we have given all the reasons why the pair (C, N) is called the Hurewicz test for Fσ sets. Since the work of Hurewicz Theorem 1 has been strengthened in many successive steps. We will discuss this in the next section. Before doing this we recall some basic definitions in descriptive set theory. Our basic reference for this is [1]. All the spaces we consider are Polish. The product spaces 2 and ω are endowed with their usual product topology, that we denote by τ2ω and τωω respectively. For two arbitrary sets A, B in a topological space X we say that A is Wadge reducible to B, which we denote as A ≤W B, if there is a continuous function f : X → X such that f−1(B) = A. This preordering gives rise to the equivalence relation A ≡W B ⇐⇒ A ≤W B ∧ B ≤W A. If