I know Klein bottle is a $2$-d smooth manifold with self intersection and has $2$-d tangent space at each point. I can image this for points where the manifold does not intersect itself. My trouble is I can't image a $2$-d tangent space at a self-intersection point.
In my view, the neighborhood of a self-intersection point locally looks like two $2$-d planes intersect. Then the tangent vectors at this point cloud be classified to be two classes; one class of vectors are tangent to one of the intersecting planes. That is, the tangent space at this point consists of two $2$-d planes. But if so, the tangent space in whole is not a vector space any more. What's wrong with my view ?
Here I adopt this definition of tangent vector:
Tangent vectors at $p$ are the derivatives of smooth paths in the manifold passing $p$.