I have been trying to find a visual transformation which is as I desire, but cannot find one. $\mathbb{R}$ is obviously homeomorphic to itself, and for any two sets of same finite cardinality it isn't to hard to find a homeomorphism which in addition is a bijection of these two sets. (By ordering the set, and then expanding the real line in certain areas and retracting it in others)
My questions is, does this property hold if the sets are of countable cardinality, in particular considering $\mathbb{Q}$ and $\mathbb{N}$.
If this result is true, it would be cool to have a constructive proof.
Homeomorphisms restrict to homeomorphisms, so if we had such a map, we would have a homeomorphism $\mathbb{Q} \cong \mathbb{N}$. But there is no singleton open set in $\mathbb{Q}$, while there are such sets in $\mathbb{N}$. So this can’t happen.