Variations on the Wadge Reducibility

The Wadge reducibility in the Baire and Cantor spaces is very important in descriptive set theory. We consider the Wadge reducibility in some other topological spaces, in particular, in the $\varphi$-spaces which are topological counterparts of the algebraic directed-complete partial orderings. It turns out that the Wadge reducibility behaves worse in most spaces than in the classical case but there exist interesting examples of spaces with a better behavior as well.