Editing Invariance of domain (section)

==Generalizations==

The domain invariance theorem may be generalized to [[manifold]]s: if <math>M</math> and <math>N</math> are topological {{mvar|n}}-manifolds without boundary and <math>f : M \to N</math> is a continuous map which is [[Locally injective function|locally one-to-one]] (meaning that every point in <math>M</math> has a [[Neighborhood (topology)|neighborhood]] such that <math>f</math> restricted to this neighborhood is injective), then <math>f</math> is an [[open map]] (meaning that <math>f(U)</math> is open in <math>N</math> whenever <math>U</math> is an open subset of <math>M</math>) and a [[local homeomorphism]].

There are also generalizations to certain types of continuous maps from a [[Banach space]] to itself.<ref>{{aut|[[Jean Leray|Leray J.]]}} Topologie des espaces abstraits de M. Banach. ''C. R. Acad. Sci. Paris'', 200 (1935)  pages 1083–1093</ref>