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

TL;DR

This paper provides an elementary proof that no continuous bijection exists between ^n and ^2 for na0a0 2, using partitioning into dense arcwise connected subsets.

Contribution

It introduces a simple method to prove the nonexistence of such bijections by partitioning ^n into dense arcwise connected subsets for na0a0 3.

Findings

No continuous bijection exists between ^n and ^2 for na0a0 2.

Partitioning ^n into dense arcwise connected subsets is possible for na0a0 3.

Elementary proof technique for nonexistence of certain continuous bijections.

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.