The open unit disk and the open unit square $(0,1) \times(0,1)$ are homeomorphic.

Question

The open unit disk in $\Bbb R^2$ centered at origin and the open unit square $(0,1) \times(0,1)$ are homeomorphic.

I have proved it in following steps:

1) open unit disk is homeomorphic to $\Bbb R^2$

2) $(-1,1)$ is homeomorphic to $\Bbb R$

3) $(-1,1)$ is homeomorphic to $(0,1)$

Thus we have a homeomorphism from $(0,1)$ to $\Bbb R$ let's say $h$.

4) Define $f : \text{open unit square} \to \Bbb R^2$ as $(x,y) \to (h(x),h(y))$

Thus combining 4 and 1 we have homeomorphism from open unit disk to open unit square.

Is the solution correct? If there is any easy method of doing it please let me know!

The 1st map is given by $x \to \frac{x}{1-\|x\|}$ and the inverse map by $x \to \frac{x}{1+\|x\|}$.

Answer 1

This looks good to me! Personally, I'd combine steps 2 and 3 and show directly that $\mathbb{R}$ is homeomorphic to $(0,1)$ but that's really personal choice.

A different way could be to construct the homeomorphism between the two spaces directly, but your approach seems nicer to me, and more intuitive.

Answer 2

It does look fine, but to be honest your step $1.$ seems to be exactly as difficult as the original problem. How did you prove that? Are you sure that you can't adapt your proof to get the required homeomorphism directly?