Finding a homeomorphismus between $\big(X=\mathbb{R}^2,\mathcal{T}(d_2)\big)$, $\big(Y=(-1,1)\times(-1,1), \mathcal{T} (d_2 \vert_{Y \times Y})\big)$

Question

$\big(X=\mathbb{R}^2,\mathcal{T}(d_2)\big)$, $\big(Y=(-1,1)\times(-1,1), \mathcal{T} (d_2 \vert_{Y \times Y})\big)$.

I need to find a homeomorphism from $X \rightarrow Y$.

My try is:

$f:\mathbb{R}^2 \rightarrow (-1,1)\times(-1,1)$

$(x,y) \mapsto \big( \frac{2}{\pi} \arctan(x), \frac{2}{\pi} \arctan(y)\big)$

$f$ is bijective and continuous because $\arctan$ is and $f^{-1}$ is continuous on $(-1,1)\times(-1,1)$. Am I done here or do I need to show more?

Answer 1

Everything is correct! There are plenty of homeomorphisms $\mathbb{R} \to (-1,1)$ which you can use instead of $\frac{2}{\pi}\arctan$.