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

Daneshpajouh, Hamid; Daneshpajouh, Hamed; Malek, Fereshte

doi:10.2140/involve.2014.7.125

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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

DOI: 10.2140/involve.2014.7.125

Abstract

We show that there is no continuous bijection from $ℝ^{n}$ onto $ℝ^{2}$ for $n \neq 2$ by an elementary method. This proof is based on showing that for any cardinal number $β \leq 2^{ℵ_{0}}$ , there is a partition of $R^{n}$ ( $n \geq 3$ ) into $β$ arcwise connected dense subsets.

Keywords

arcwise connected, dense subset, homeomorphism

Mathematical Subject Classification 2010

Primary: 54-XX

Secondary: 54CXX

Milestones

Received: 3 June 2012

Revised: 27 November 2012

Accepted: 1 December 2012

Published: 16 November 2013

Communicated by Joel Foisy

Authors

Hamid Reza Daneshpajouh
School of Mathematics Institute for Research in Fundamental Sciences P.O. Box 19395-5746 Tehran Iran

Hamed Daneshpajouh
Department of Mathematical Sciences Sharif University of Technology P.O. Box 11155-9415 Tehran Iran

Fereshte Malek
Department of Mathematics Faculty of Science K. N. Toosi University of Technology P.O. Box 16315-1618 Tehran Iran

Vol. 7, No. 2, 2014