Given any bijection $\varphi$ between the irrationals and $\omega^\omega$, and a subset $A \subseteq \mathbb{R} \smallsetminus \mathbb{Q}$ of size $\mathfrak{d}$ , under which properties $\varphi(A)$ is dominating?
What about the question in the title?
I guess we should assume $\mathfrak{d} < \mathfrak{c}$.



Is there a subset of irrationals of size $\mathfrak{d}$ whose image, under any bijection to the Baire space, remains dominating?
For the title question, the answer is no, when d is less than c, because you could map your set into the 2-valued functions and map the complement to the rest. So the image would not be dominating.


