Given Baire class $1$ function $f:\omega^\omega\rightarrow \omega^\omega$ I want to prove that for every non empty $U_1,U_2$ open sets with $U_1\cap U_2 = \emptyset$, there exists a $\boldsymbol{\Delta}_2^0$ set $B$ s.t. $f^{-1}(U_1) \subseteq B$ and $B\cap f^{-1}(U_2) = \emptyset$, i.e. $B$ separates $f^{-1}(U_1)$ from $f^{-1}(U_2)$.
I know that in general in zero dimensional spaces the bolface classes $\boldsymbol{\Sigma}_2^0, \boldsymbol{\Sigma}_1^0$ do not have the separation property.
Any idea?
What if I worked with a Baire class $3$ function and asked for the existence of a separating $\boldsymbol{\Delta}_4^0$ set?
Thanks!