As regards Q (your first remark), it is true that all countable metrisable spaces without isolated points are homeomorphic to Q. If you want to omit metrisable, replace it by T_3 and second countable. One then notes that a dense subset of R^n doesn't have isolated points, and is metrisable.

As regards $\mathbb Q$ (your first remark), it is true that all countable metrisable spaces without isolated points are homeomorphic to $\mathbb Q$. If you want to omit metrisable, replace it by $\mathrm T_3$ and second countable. One then notes that a dense subset of $\mathbb R^n$ doesn't have isolated points, and is metrisable.