Compact covering and game determinacy

All spaces are separable and metrizable. Suppose that the continuous and onto mapping t́f : X → Y is compact covering. Under the axiom of ∑11-determinacy, we prove that t́f is inductively perfect whenever X is Borel, and it follows then that Y is also Borel. Under the axiom ℵ1L = ℵ1 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.