#### Vol. 7, No. 2, 2014

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

### Hamid Reza Daneshpajouh, Hamed Daneshpajouh and Fereshte Malek

Vol. 7 (2014), No. 2, 125–127
##### 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.

##### Keywords
arcwise connected, dense subset, homeomorphism
Primary: 54-XX
Secondary: 54CXX