TITLE: Injective maps $\mathbb{R}^{n} \to \mathbb{R}^{m}$
QUESTION [7 upvotes]: Let $f: \mathbb{R}^{n} \to \mathbb{R}^{m}$ be an injection for $n>m$. Can $f$ be continuous? Why?
I got this question in mind when I was trying to find a continuous map from $\mathbb{R}^{2}$ to $\mathbb{R}$.

REPLY [2 votes]: I taught an introductory topology course this last autumn where I covered this theorem from an elementary point of view. The argument just uses the Brouwer fixed point theorem (which itself has a proof via Stokes' theorem which is readily accessible to students with several variable calculus) plus elementary point set topology. In particular no homology or Jordan--Brouwer separation theorems are used. The treatment was based upon that of Hurewicz & Wallman but was also inspired by Larry Guth's ICM-2010 presentation in Hyderabad.
If you're interested the notes are available on 
http://www.maths.ed.ac.uk/uploads/assets/32_section6.pdf