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

Abstract

In this article it is shown that there is no continuous bijection from $\mathbb{R}^n$ onto $\mathbb{R}^2$ for $n\neq 2$ by an elementary method. This proof is based on showing that for any cardinal number $\beta\leq 2^{\aleph_0}$, there is a partition of $R^n$ ($n\geq 3$) into $\beta$ arcwise connected dense subsets.