Let $\mathbb P$ denote the space of irrationals. Is there a continuous bijection (one-to-one and onto) $f:\mathbb P\to \mathbb Q ^\omega$ that maps each closed subset of $\mathbb P$ to a $G_\delta$-subset of $\mathbb Q ^\omega$?

**Remark 1**. Suppose that $f:\mathbb P\to \mathbb Q ^\omega$ is a continuous bijection mapping closed sets to $G_{\delta}$ sets. Then $f^{-1}:\mathbb Q^\omega\to \mathbb P$ is a *Baire class $1$ functions*, i.e. $(f^{-1})^{-1}(U)=f(U)$ is an $F_{\sigma}$-subset of $\mathbb Q ^\omega$ for every open set $U\subseteq \mathbb P$. By Theorem 4.1 in the reference below, either there are countably many sets $X_n\subseteq \mathbb Q ^\omega$ such that $\mathbb Q ^\omega=\bigcup \{X_n:n<\omega\}$ and $f^{-1}\restriction X_n$ is continuous, or $f^{-1}$ contains Pawlikowski's function $P:(\omega+1)^\omega\to \omega^\omega$.

**Remark 2**. While trying to solve this problem, I discovered an example involving complete Erdos space $\mathfrak E_c$. There is a continuous bijection $f:\mathfrak E_c^\omega\to \mathbb Q ^\omega$ which maps closed sets to $G_{\delta}$ sets and such that $f^{-1}$ is not a countable union of continuous functions. So by Theorem 4.1 it must contain $P$. I proved that all similar examples, including the one in @Arno's answer also must contain $P$.

My feeling now is that my question probably has a positive answer, although the zero-dimensionality of $\mathbb P$ makes things interesting.

*Solecki, Sławomir*, **Decomposing Borel sets and functions and the structure of Baire class 1 functions**, J. Am. Math. Soc. 11, No. 3, 521-550 (1998). ZBL0899.03034.