I would like to receive your help with this question.
The tangent cones are the generalizations of the tangent spaces. If we take the fiber bundle $TM$ we have for it a structure of a differentiable manifold. Can we do the same with the tangent cones $C_pM$ if we considerer the union of al these spaces in the same way that we the tangent bundle?
Actually I'm interested in the topological structure of that object and if we considerer such space as a topological space, what would be its base of sub-base.