Among other results we prove that the topological statement “Any compact covering mapping between two Π⁰₃ spaces is inductively perfect” is equivalent to the set-theoretical statement "$\forall \alpha \in {\omega }^{\omega },\omega {₁}^{L\left(\alpha \right)}<\omega ₁$"; and that the statement “Any compact covering mapping between two coanalytic spaces is inductively perfect” is equivalent to “Analytic Determinacy”. We also prove that these statements are connected to some regularity properties of coanalytic cofinal sets in (X), the hyperspace of all compact subsets of a Borel...