How to find a homeomorphism between $X_{1}=(3,7)\subseteq \mathbb{R}$ and $X_{2}=\{(x,y)\in\mathbb{R}^2:y^3=x^2\}$

Question

Let $X_{1}=(3,7)\subseteq \mathbb{R}$ and $X_{2}=\{(x,y)\in\mathbb{R}^2:y^3=x^2\}$. Consider $X_1$ and $X_2$ as metric subspaces of $\mathbb{R}$ and $\mathbb{R}^2$ (with the Euclidean metric) respectively.

I want to show that there is a homeomorphism between $X_1$ and $X_2$ but I'm really struggling to visualise a continuous, bijective function between the two, since the graph of $X_2$ goes beyond the small domain of $X_1$.

I'd appreciate any help.

Answer 1

Two steps:

• Make a homeomorphism between $(3,7)$ and $\mathbb R$. $x\mapsto\tan\frac{(x-5)\pi}{4}$ would do the job:
  • We subtract $5$ to get a map to $(-2,2)$,
  • Multiply by $\pi$ to get to $(-2\pi,2\pi)$,
  • Divide by $4$ to get to $(-\pi/2,\pi/2)$, and
  • Use $\tan$ to map to the whole $\mathbb R$.
• Make a homeomorphism between $\mathbb R$ and your graph: $x\mapsto (x,\sqrt[3]{x^2})$ can work.

Now all we need is to compose those two.