Local diffeomorphism

A smooth map $f : M \to N$ between two smooth manifolds $M$ and $N$ is called a local diffeomorphism if for every point $p \in M$ , there exists an open neighborhood $U$ of $p$ in $M$ and an open neighborhood $V$ of $f (p)$ in $N$ such that the restriction of $f$ to $U$ ,

f |_{U} : U \to V,

is a diffeomorphism.