Invariance of Domain

The topological theorem stating that an injective continuous map between open subsets of Euclidean n-space is an open embedding; in particular, the image of an open set in R^n under such a map is open and the map is a homeomorphism onto its image.

A topological principle (Brouwer's Invariance of Domain) that a continuous injective map between n-dimensional Euclidean manifolds (or between open subsets of R^n) is an open embedding; equivalently, a continuous injective map from an open set of R^n into R^n has open image. A key consequence is that Euclidean spaces R^m and R^n are not homeomorphic when m ≠ n.