Consider a diffeomorphism $f$ between two manifolds $M$ and $N$ (I refer only to the set of the manifold, the smoothness has not been specified). I know a diffeomorphism has the following properties:
- $f$ is continuous
- $f$ f is smooth
- $f$ is a bijection
- $f^{-1}$ is smooth
Is my definition fully correct? And does being a diffeomorphism make $f$ an open map?
Best Answer
An open map is just a function that maps an open set to an open set, whereas a continuous map satisfies that the preimage of an open set is an open set. Since a diffeomorphism $f$ is a continuous bijection, $f^{-1}$ is continuous. It follows that $f$ is an open map.
Since every homeomorphism is a continuous bijection, it is always an open map.