  We prove some results on the Wadge order on the space of sets of natural
numbers endowed with Scott topology, and more generally, on omega-continuous
domains. Using alternating decreasing chains we characterize the property of
Wadge hardness for the classes of the Hausdorff difference hierarchy (iterated
differences of open sets). A similar characterization holds for Wadge
one-to-one and finite-to-one completeness. We consider the same questions for
the effectivization of the Wadge relation. We also show that for the space of
sets of natural numbers endowed with the Scott topology, in each class of the
Hausdorff difference hierarchy there are two strictly increasing chains of
Wadge degrees of sets properly in that class. The length of these chains is the
rank of the considered class, and each element in one chain is incomparable
with all the elements in the other chain.
