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 n onto 2 for n2 by an elementary method. This proof is based on showing that for any cardinal number β 20, there is a partition of Rn (n 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