bijective measurable map existence

Question

Does there exist bijective measurable maps between $\mathbb{R}$ and $\mathbb{R}^n$?

If so, could you give me an example of that?

Thank you.

Answer 1

Polish space is a topological space that is isomorphic to a complete separable metric space, for example $\Bbb R^n$ for any $n\in \Bbb N$. For the proof of the following fact, see e.g. here.

Any uncountable Polish space is Borel isomorphic (there exists a bimeasurable bijection) to the space of real numbers $\Bbb R$ with standard topology.