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

Freshteh Malek; Hamed Daneshpajouh; Hamidreza Daneshpajouh; Johannes Hahn

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

General Topology 2010-03-16 v2

Authors: Freshteh Malek , Hamed Daneshpajouh , Hamidreza Daneshpajouh , Johannes Hahn

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.

Keywords

continuous bijection nonexistence arcwise connected dense subsets cardinal number elementary method

Cite

@article{arxiv.1003.1467,
  title  = {An interesting proof of the nonexistence continuous bijection between $\mathbb{R}^n$ and $\mathbb{R}^2$ for $n\neq 2$}},
  author = {Freshteh Malek and Hamed Daneshpajouh and Hamidreza Daneshpajouh and Johannes Hahn},
  journal= {arXiv preprint arXiv:1003.1467},
  year   = {2010}
}

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3 pages

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

Abstract

Keywords

Cite

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