Classifying Countable Boolean Terms

We deal with the Borel and difference hierarchies in the space $P\omega$ of all subsets of $\omega$ endowed with the Scott topology. (The spaces $P\omega$ and $2^\omega$ coincide set-theoretically but differ topologically.) We look at the Wadge reducibility in $P\omega$. The results obtained are applied to the problem of characterizing $\omega_1$ – terms $t$ which satisfy $\mathcal C =t({\boldsymbol\Sigma}^0_1)$ for a given Borel – Wadge class $\mathcal C$. We give its solution for some levels of the Wadge hierarchy, in particular, all levels of the Hausdorff difference hierarchy. Finally, we come up with a discussion of some relevant facts and open questions.