Exercise on $\boldsymbol{\Delta}_2^0$ sets in the Baire space $\omega^\omega$

Question

I want to prove the following statement:

Given a set $A\subseteq \omega^\omega$, if there exists a continuous function $g: 2^\omega \rightarrow \omega^\omega$ s.t. $f(\{z \in 2^\omega \mid \exists n \forall m\ge n \ (z_m = 0)\}) \subseteq A$ and $f(\{z \in 2^\omega \mid \exists n \forall m\ge n \ (z_m = 1)\}) \cap A = \emptyset$ then $A$ is not $\boldsymbol{\Delta}_2^0$

I tried to prove that there exists a continuous reduction of $\{z \in 2^\omega \mid \exists n \forall m\ge n \ (z_m = 0)\}$, which is $\boldsymbol{\Sigma}_2^0$-complete, onto $A$, but without much success. Any hint?

Thanks

Answer 1

Consider the set $B=f^{-1}(A)\subseteq 2^\omega$, separating the set $\{z\mid \text{cofinitely }0\}$ from the set $\{z\mid \text{cofinitely }1\}$. It suffices to show that $B$ cannot be $\mathbf{\Delta^0_2}$. Assume that $B$ is, and we find that $B$ is both dense $G_\delta$ and codense $F_\sigma$, meaning that $B$ is both comeager and meager. This violates the Baire category theorem.