Find a bijection from the closed disk to the open square This is a slightly specific question but I'm not sure how to even begin or go about doing it:
Find a bijection from a closed disk:
$D=\{(x,y)\in\mathbb{R}^2|x^2+y^2\leq 1\}$
To the open square:
$S = (0,1) \times (0,1)=\{(x,y)\in \mathbb{R}^2|0<x<1, 0<y<1\}$
 A: You can do this in a sequence of steps. The only tricky one is the first.
1) Find a bijection from the closed unit disk to the open unit disk. 
2) Find a bijection from the open unit disk to the square $(-1,1) \times (-1,1)$
3) Find a bijection from the square $(-1,1) \times (-1,1)$ to the square $(0,1) \times (0,1)$.
One bijection from the closed unit disk to the open unit disk is
$$ f(x,y) = \left\{ \begin{array}{lc} (x/2,y/2) & x^2 + y^2 = 4^{-n},\ n \in \{0,1,2,\ldots\} \\
(x,y) & \text{otherwise} \end{array} \right.$$
A: *

*If we define $\varphi$ through $\varphi(0,0)=(0,0)$ and $\varphi(x,y)=\frac{\max(|x|,|y|)}{\sqrt{x^2+y^2}}(x,y)$ then $\varphi$ maps the closed unit disk into the closed square $\max(|x|,|y|)\leq 1$ in a bijective way;

*There is a bijective map $\gamma$ from $[-1,1]$ to $(-1,1)$: for instance, it is enough to enumerate the rational points of $[-1,1]$ as $q_0=-1,q_1=1,\ldots$, then map $q_n$ into $q_{n+2}$;

*Let $\eta(x,y) = \left(\frac{\gamma(x)+1}{2},\frac{\gamma(y)+1}{2}\right)$. Then $\eta\circ\varphi$ maps $x^2+y^2\leq 1$ into $(0,1)\times(0,1)$ in a bijective way.


Of course there are some issues in dealing with continuous bijections since $x^2+y^2\leq 1$ is closed while $(0,1)\times(0,1)$ is open in the Euclidean topology.
