Now there's even more structure that can be extracted from the charts.

Choose a point \$$s \in M\$$, and let \$$C(s) \in \mathbb{R^n}\$$ be its coordinate representation.

In addition to the tangent space \$$\mathit{Tangent(s)}\$$ at \$$s\$$ in the manifold itself, there is also a tangent space \$$\mathit{Tangent(C(s))}\$$ around \$$C(s) \in \mathbb{R^n}\$$.

That's because \$$\mathbb{R^n}\$$ is itself a simple kind of manifold, at so at every point in \$$\mathbb{R^n}\$$ there is a vector space of tangent vectors at that point.

Now that tangent space \$$\mathit{Tangent}(C(s))\$$ turns out to be just a copy of \$$\mathit{R^n}\$$, but in principle we still visualize as a local copy of \$$\mathbb{R^n}\$$ at the point \$$C(s))\$$.