How to prove $\mathbb{R}\times \mathbb{R}$ is equipotent (i.e. there is a 1-1 map between the two) to $\mathbb{R}$?
The way to prove $\mathbb{N}\times \mathbb{N}$ is equipotent to is to $\mathbb{N}$ is to construct a function of two independent variables and one dependent variable, e.g. $g(m,n)=(m+n)^2+n$, and prove that it is bijective.
How can we construct such a function for $\mathbb{R}\times \mathbb{R}\to\mathbb{R}$, or $\mathbb{C}\to\mathbb{R}$?
It seems that we can use fractals to cover a two dimensional plane with a line. But what exactly are such fractals? And how can we write them with functional expression?
