[Math] How to show that the right half plane of $\mathbb{R}^2$ is homeomorphic to the open unit disk

Question

I am trying to show that $\{(x,y)\in\mathbb{R}^2:x>0\}$ is homeomorphic to the disk $\{(x,y)\in\mathbb{R}^2:x^2+y^2<1\}$. I know that the disk is homeomorphic to the whole plane. A homeomorphism from $\mathbb{R}^2$ to the disk could be $f(x,y)=(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2})$. I know that being homeomorphic is an equivalence relation. So I was looking for a homeomorphism between the half plane and the whole plane, but I couldn't find one. Any idea?

Answer 1

Hint: $x \mapsto e^x$ is a homeomorphism between $\mathbb{R}$ and $(0, \infty)$