An interesting proof of the nonexistence of a continuous bijection between $\mathbb{R}^n$ and $\mathbb{R}^2$ for $n\neq 2$

Abstract

We show that there is no continuous bijection from ${ℝ}^n$ onto ${ℝ}^2$ for $n\ne 2$ by an elementary method. This proof is based on showing that for any cardinal number $\beta \le 2^{{\aleph }_0}$, there is a partition of $R^n$ ($n\ge 3$) into $\beta$ arcwise connected dense subsets.