I'm quite new with manifold. Let $M$ a manifold of dimension $2$ and let $\Sigma\subset M$ a curve (i.e. a sub-manifold of dimension $1$). They say in my lecture : We choose a coordinate system s.t. $\Sigma=\{(x,y)\in \mathbb R^2\mid x=0 \}$.
Q1) I'm not sure how this is possible. How can I prove that such a coordinate system exist ? Alors, as written suggest that $\Sigma\subset \mathbb R^2$, but since $M$ is not $\mathbb R^2$ a priori, it may have a mistakes, no ?
Q2) If indeed there is a mistakes, suppose $M=\mathbb R^2$. Then, how is it possible that $\Sigma=\{(x,y)\mid x=0\}$ since we don't know the curve a priori ?