Bijection between $\mathbb R$ and $\mathbb R^N$

Question

Hi, could you give me an example of a bijection between $\mathbb R^T$ and $\mathbb R$ with $T$ being a positive integer which is as regular as possible like $C^{\infty}$ or Lipschitz continuous at least?

Thanks a lot!

Answer 1

There is no injective continuous map from $\mathbb{R}^n$ to $\mathbb{R}$ for $n \geq 2$. Indeed, by contradiction suppose there exists such a map $\phi : \mathbb{R}^n \to \mathbb{R}$; then $\phi(\mathbb{R}^n \backslash\{0\})=\mathbb{R} \backslash \{\phi(0)\}$ shoud be connected.

For examples of bijections, you can see this excellent answer.

Answer 2

See the related question

Is it true that a space-filling curve cannot be injective everywhere?

That is, any map from $\mathbb R$ to $\mathbb R^n$ (with $n \in \mathbb Z$ and $n>1$) cannot be both continuous and a bijection.