Boole vs Wadge: comparing basic tools of descriptive set theory

We systematically compare omega-Boolean classes (obtained from open sets (or other classes) by applying omega-Boolean operations), the reducibility by continuous functions (known as Wadge reducibility), and the recent extension of Wadge hierarchy to non-zero-dimensional spaces. E.g., we complement the result of W. Wadge that the collection of non-self-dual levels of his hierarchy coincides with the collection of classes generated by Borel omega-ary Boolean operations from the open sets in the Baire space. Namely, we characterize the operations, which generate any given level in this way, in terms of the Wadge hierarchy in the Scott domain. As a corollary we deduce the non-collapse of the latter hierarchy. Also, the effective version of this topic is discussed.