Preservation of complete metrizability by covering maps

A map f:X→Yf:X→Y is said to be w-covering if for every ordinal α   and every compact S⊂YS⊂Y such that the α  -th Cantor derivative (S)α(S)α of S   is a singleton {y}{y} there is a compact subset E of X   such that f(E)⊂Sf(E)⊂S and (f(E))α={y}(f(E))α={y}.We investigate the preservation of completely metrizability by various kinds of w-covering maps.