Preservation of the Borel class under countable-compact-covering mappings

The aim of this note is to prove the following result: “Assume that X is a metric Borel space of class ξ, that f : X → Y is continuous, that every fiber f −1 ( y ) is complete and that every countable compact subset of Y is the image by f of some compact subset of X. Then Y is Borel and moreover of class ξ”. We give also an extension to the case where the fibers are only assumed to be Polish.