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 \(n=m\) (Brouver, 1910 ) .

Remarks
  • Invariance of dimension implies that \(\mathbb{R}^n\) and \(\mathbb{R}^m\) are not homeomorphic if \(m\neq n\).
  • Invariance of dimension implies that a topological space cannot be both a \(n\)- and a \(m\)-manifold with \(n\neq m\).
  • Indeed, this result is easier to prove for an diffeomorphism. Because then the differential of the diffeomorphism is invertible, which can be seen applying the chain rule. This implies \(m=n\).