Remember that we say for $\alpha,\beta$ in $\omega^\omega$, that $\alpha\leq_T \beta$ if $\alpha$ is recursive in $\beta$.
Is $\leq_T$ a $\Sigma^1_1$ set, as a subset of $\omega^\omega\times \omega^\omega$?
Basically; I need this to prove that if $Q(x,y)$ is $\Pi^1_1$ then
\begin{equation} P(x) \Leftrightarrow \forall y\; (y\leq_Tx \Rightarrow Q(x,y)) \end{equation} is $\Pi^1_1$. So maybe the question above might have a negative answer; but another result such as a kind of parametrization for $\Sigma^0_1(x)$ points might work, so then the question will be if there is such a parametrization.
(All clases are lightfaced)