In our Riemannian geometry class we were asked to verify the following:
Let $M$ be a $k$-dimensional manifold in $\mathbb{R}^n$ and let $\varphi:U\subset\mathbb{R}^k\to M\subset\mathbb{R}^n$ be a $C^1$ coordinate chart with $U$ open. Then if \varphi'(x) has full rank $\forall x\in U$ we have $\varphi(U)$ is open in $M$ (which I'm assuming means $\varphi(U)=M\cap V$ for some $V\subset\mathbb{R}^n$ , $V$ open?)
Any points of advice to get me started in the right direction? I not sure how I should approach proving something is open in the subset-topology sense.