invariance of dimension

The following non-trivial result was proven by Brouwer [1] around 1910 [2].

Theorem (Invariance of dimension) Suppose $U$ and $V$ are open subsets of $\mathbb{R}^{n}$ respectively $\mathbb{R}^{m}$ . If $U$ and $V$ are non-empty and homeomorphic, then $n=m$ .

References

1 The MacTutor History of Mathematics archive, http://www-gap.dcs.st-and.ac.uk/ history/Mathematicians/Brouwer.htmlentry on Luitzen Egbertus Jan Brouwer
2 A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. Also available http://www.math.cornell.edu/ hatcher/AT/ATpage.htmlonline.