Page "Euclidean space" Paragraph 38

An important result on the topology of R < sup > n </ sup >, that is far from superficial, is Brouwer's invariance of domain.

Any subset of R < sup > n </ sup > ( with its subspace topology ) that is homeomorphic to another open subset of R < sup > n </ sup > is itself open.

An immediate consequence of this is that R < sup > m </ sup > is not homeomorphic to R < sup > n </ sup > if m ≠ n — an intuitively " obvious " result which is nonetheless difficult to prove.