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

Abstract

In this article it is shown that there is no continuous bijection from ${\mathbb{R}}^n$ onto ${\mathbb{R}}^2$ 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.

3 pages