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

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

2010-03-07

arxiv.org/abs/1003.1467v2

Mathematics

General Topology

3 pages

Scientific paper

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.

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