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