We can identify $S^2$ inside of ${\mathbb{R}}^3$ as $\delta B_1(0)$. There is a canonical homeomorphism called the stereographic projection:

This is constructed in the following way:

1. We distinguish the point $(0,0,1)$, which is the point we will be projecting from. We we identify the $x-y$ axis with 1;;$\mathbb{C}$.
2. Set $\sigma ((0,0,1))=\mathrm{\infty }$. For every other point $(\alpha ,\beta ,\gamma )$ on the sphere, we draw 1;;the line connecting $(0,0,1)$ with $(\alpha ,\beta ,\gamma )$, and observe the place where it crosses the complex plane (It must at some point because the line is not parallel to the plane). One can check that this map is continuous.
3. We would also like to determine the inverse map:${\sigma }^{-1}:\hat{\mathbb{C}}\to S^2$. If we do the computations, we find that the inverse map has the form ${\sigma }^{-1}(x+iy)=\frac{2}{1+|z{|}^2}(x,y,|z{|}^2-1)$ (sending $\mathrm{\infty }$ to the north pole), and we note taht this map is also continuous.