looked through this forum and didn't find an understandable solution for this problem, would love your help.
A universal-$\Sigma_\alpha^0$ set $U$ in the Baire space would be a $\Sigma_\alpha^0$-Borel set under $\omega^\omega\times\omega^\omega$, such that for any $\Sigma_\alpha^0$-Borel set B under $\omega^\omega$ there is $e\in\omega^\omega$ such that $B=\{x: (x,e)\in U\}$.
I think I might've not completely understood the second term for a universal set.
In the induction base $\alpha=1$ I constructed an open set of $(x,y) $such that there is an $n$ such that $x(n)=y(n)$.
In the induction step I tried to use the $e$ that coded the universal set $U_\alpha$ to create $U_{\alpha+1}$. Didn't quite get this part and how to execute it, and why exactly it creates a universal $\Sigma_{\alpha+1}^0$ set?
Anyways would like clarifications.
