Collapsing the Unit Ball in $\mathbb{R}^n$

Question

For a homework question, I need to prove that $\mathbb{R}^n/\overline{B_1(0)}$ is homeomorphic to $\mathbb{R}^n$, where $\overline{B_1(0)}$ is the unit ball centered at the origin. Clearly, the inclusion map $\mathbb{R}^n\backslash\overline{B_1(0)} \hookrightarrow \ \mathbb{R}^n/\overline{B_1(0)}$ is a homeomorphism onto an open set; can I use this fact somehow and smoothly extend this map to produce a homeomorphism between the total spaces?