If $U$ is an open subset of the manifold $X$, check that $$T_x(U) = T_x(X) \text{ for } x \in U.$$
I only proved when $U$ is an open subset of the manifold $X$, which is not true for submanifolds of $X$ in general right? My thought is considering the tangent space for $\mathbb{R}^3$ is $\mathbb{R}^3$, but the tangent space for its submanifold $\mathbb{R}^2$ is $\mathbb{R}^2$.
More generally, an open subset of the manifold $X$ is a submanifold, but not all submanifolds are an open subset of the manifold $X$ - some are reduced dimensions, and some more out there that I don't know. - correct?
