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1	Homeomorphism
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Invariance Of Dimension

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

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

References

1: The MacTutor History of Mathematics archive, entry on Luitzen Egbertus Jan Brouwer
2: A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. Also available online.