Differential Geometry: Lecture 12 part 5: TpM the tangent space

Finally, the end is near. I define the tangent space to M as the set of all possible tangents to curves on M. The tangents to curves naturally fall in TpR3 which we discussed in earlier lectures. Naturally, the tangent space to the surface M is seen to be spanned by the partial velocities of the chart X. In particular, we find the tangent space forms a two-dimensional subspace of the total tangent space at each point along M. Moreover, vector fields along M are introduced, in fact they are sections of the tangent bundle over M (which we have not yet introduced). Normal vector fields to M also introduced. We have much more to say about those later.

[stellamn]

Sorry but we learned that f must map from an open set of M to R^n and be an homeomorphism in order to show that M is locally Euclidean... so why now you restriction to 1 dimension?