I know that $\mathbb{R}^2$ and $\mathbb{R}^2\setminus\{(0,0)\}$ are not homeomorphic. (For examle $\pi_1(\mathbb{R}^2)=\{e\}$, but $\pi_1(\mathbb{R}^2\setminus\{(0,0)\})=\mathbb{Z}$).
But what can be said about $\mathbb{Q}^2$ and $\mathbb{Q}^2\setminus\{(0,0)\}$? Is there a homeomorphism?