Let $U, V \subset \mathbb{R}^n$ two subsets of $\mathbb{R}^n$ such that $U$ is open. Let $\gamma \in C^{\infty}(U, \mathbb{R}^n)$ an injective continuous map with the property that $\gamma(U) =V $ and $\gamma: U \to V$ is a diffeomorphism.
Is then $V$ open in $\mathbb{R}^n$? If yes, why?
Consideration: Surely, since $\gamma$ is a diffeomorphsim in $C^{\infty}(U, V)$, it is there a homeomorphism and therefore there an open map. But here we consider $V= \gamma(U)$ as a subset of $\mathbb{R}^n$. How to see that $\gamma(U)$ is open in $\mathbb{R}^n$?