Is it okay to do that with proving $\mathbb R^2$ and $\mathbb R^2 \setminus \left\{0\right\}$ are homeomorphic?
Solution 1:
It is not true that ${\mathbb{R}^2 - (0,0)}$ and ${\mathbb{R}^2}$ are homeomorphic. In fact, they don't even have the same homotopy type, since ${\mathbb{R}^2 - (0,0)\simeq S^1}$, we have $$ \pi_1(\mathbb{R}^2 - (0,0)) = \pi_1(S^1) = \mathbb{Z},\text{ but }\pi_1(\mathbb{R}^2) \equiv 0 $$