Abstract All spaces are separable and metrizable. Suppose that the continuous and onto mapping t ́ f : X → Y is compact covering. Under the axiom of ∑ 1 1-determinacy, we prove that t́f is inductively perfect whenever X is Borel, and it follows then that Y is also Borel. Under the axiom ℵ 1 L = ℵ 1 we construct examples showing that the conclusion might fail if “ X is Borel” is replaced by “ X is coanalytic”. If we suppose that both X and Y are Borel, then we prove (in ZFC) the weaker conclusion that t́f has a Borel (in fact a Baire-1) section g : Y → X. We also prove (in ZFC) that if we suppose only X to be Borel but of some “low” class, then Y is also Borel of the same class. Other related problems are discussed.