An interesting proof of the nonexistence of a continuous bijection between ℝn and ℝ2 for n ≠ 2

Hamid Reza Daneshpajouh; Hamed Daneshpajouh; Fereshte Malek

doi:10.2140/involve.2014.7.125

An interesting proof of the nonexistence of a continuous bijection between ℝⁿ and ℝ² for n ≠ 2

Hamid Reza Daneshpajouh, Hamed Daneshpajouh, Fereshte Malek

Research output: Journal Publication › Article › peer-review

Abstract

We show that there is no continuous bijection from ℝⁿ onto ℝ² for n ≠ 2 by an elementary method. This proof is based on showing that for any cardinal number β ≤ 2^{א 0}, there is a partition of Rⁿ (n ≥ 3) into arcwise connected dense subsets.

Original language	English
Pages (from-to)	125-127
Number of pages	3
Journal	Involve
Volume	7
Issue number	2
DOIs	https://doi.org/10.2140/involve.2014.7.125
Publication status	Published - 2014
Externally published	Yes

Keywords

arcwise connected
dense subset
homeomorphism

ASJC Scopus subject areas

General Mathematics

Access to Document

10.2140/involve.2014.7.125

Cite this

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abstract = "We show that there is no continuous bijection from ℝn onto ℝ2 for n ≠ 2 by an elementary method. This proof is based on showing that for any cardinal number β ≤ 2א 0, there is a partition of Rn (n ≥ 3) into arcwise connected dense subsets.",

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