I like differential geometry and I want to know if a differentiable manifold implies unique tangent space at every point. I have searched but the definition I have found of differential manifold is that differentiable manifold is a topological manifold in which we construct an atlas that covers the entire manifold and the transition functions in the intersection of the coordinate systems of the atlas are differentiable. But this doesn't say anything about tangent spaces. Whatever I have read about differential geometry, tangent spaces are defined always after the definition of differentiable manifold, so it seems tangent spaces are not need to define what a differentiable manifold is and for a topological manifold to be a differentiable manifold is only need that we construct an atlas with the characteristics I said before. But then if you have for example, a 1 dimensional topological manifold that is a real line with one point that is not smooth, because it has the form of a peak, there is no unique tangent space at this point, although there is in the other points. Then we construct an atlas in which the transition functions are differentiable at this point just because we can define the coordinate systems in a way that the transition functions from one coordinate to another is differentiable. Although the manifold has no unique tangent space in this point. I always believed that differentiability at manifold level implies unique tangent space and that the manifold doesn't care about coordinate system differentiability. But the definition of differentiable manifold is based on differentiability in coordinate systems. So if we can define an atlas that is smooth but the real manifold is not, Therefore, differentiable manifold doesn't implies unique tangent space. Am I missing something?
Sorry for my bad expression but english is not my mother tongue and I tried to express the better I can.