In string theory there are more than 3+1 dimensions of the space-time, but this is only at very very small "scales", and the concept of dimension used in physics are the one of manifolds because this set the dimension of n-Euclidean space $E^n$ as n. And this is a topological dimension and all the tpological dimensions that i know satisfy the subspace theorem that say that every subspace have dimension less or equal of the space. In some occasions the physicist describe this "phenomena" as the dimensions are "curled-up" and many persons (in the web) like in this question "Curled-up dimensions"? think that the extra "curled-up" dimensions are the dimensions of the fibre a fibre budle but i'm understand this interpretation as that the perceivable space-time of 3+1 dimensions is the base space and the entire space-time is the fibre budle with some fibre. But this interpretation seems to me that not explain the "small scale" extra dimensions, but extra dimensions as in a embedding of the peceivable space time in the entire space-time and that thi is globally with more dimension and not locally. In any case as a analogy sometimes physicists compare a torus and that if you go away from it, it seems to be a circle, and with this i reminded coarse geometry, wich intuitivley is about the global properties of a "space" and that "look the same from afar" is an example.
is perhaps the curled dimension is about coarse geometry? or is topological? and how? or what type of dimesion is and how can a type of "space" have more dimensions in a more small "portion" of the space than in a bigger one?
Thanks in advance.