i'm studying the Wadge hierarchy on Baire space and Cantor space. I'm asking whether or not the $\Delta^0_2$ sets form a unique degree in these spaces and why the $\Sigma^0_2$-true (i.e. not polish) subsets of $2^\omega$ form a unique degree. Moreover, which results for $\omega^\omega$ can i say also for $2^\omega$?
I'll thanks anyone let me know more about the relation between wadge hierarchy and on these spaces and their boldface pointclasses.
I'll answer about the boldface versions.
The ambiguous class $\boldsymbol\Delta_2^0$ comprises $\omega_1$ levels of the Wadge hierarchy (see Kecrhis' book on descriptive set theory, Exercise 21.16, and the many references therein).
Sets in $\boldsymbol\Sigma_\alpha^0 \setminus \boldsymbol\Pi_\alpha^0$ are complete, and from this it follows that they are all inter-reducible, and hence in a unique degree.
Finally, all the results hold uniformly of all zero dimensional Polish spaces, in particular $A^\omega$ for any countable $A$.
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