Degree of a Euclidean homeomorphism ($\mathrm{deg}(f:{\mathbb{R}}^n\to {\mathbb{R}}^n)$)

A Euclidean homeomorphism is a proper map and extends to its one-point compactification $S^n$. Its degree is the degree of a continuous mapping of this extension, and belongs to $\{1,-1\}$. A linear isomorphism has degree equal to the sign of its determinant.