Prove that if $T$ is one-to-one on $D$, then the set $T(D)$ is open

Question

Let $f$ and $g$ have continuous first-order partial derivatives on an open set $ D\subseteq\mathbf{R}^2 $ and let $T :D \to \mathbf{R}^2 $ be defined by

$ T(u,v)=(f(u,v),g(u,v)). $

Prove that if $ T $ is one-to-one on $D$, then the set $T(D)$ is open.

I need help ; I try to apply the inverse function theorem, but I can't see any connection between the injectivity and Jacobian matrix of $T$. Do i need to notice something completely different?