Inspired from the fact that the postulates of Euclid describe uniquely the Euclidean space as a riemannian manifold (in the next sense: consider the "multiverse" arisen by consider the the geometries that fulfill the first 4 postulates, then the unique geometries that too fulfill the fifth postulate are the euclidean planes with other assuptions as the possibility or impossibility of the quadrature of the circle.), I question the follows: Suppose we have a theory, then there exist a theory for each object that characterize it (i.e. describes uniquely the object, absolutely, i.e. in the totality of the "multiverse" and not only as in the motivation; in certain level of the "multiverse")? . For example if we have the theory of topological spaces, there is a theory that only talks about the circle, there is a theory that only talks abut the Klein bottle, and so on?. And we can characterize absolutely the euclidean plane?
In the other hand the euclidean space have a rich amount of structures that make it a space, and there is another question (independent of the above question), there is a notion that captures the intuitive idea of being able to "accept" different structures?
If the answer of my first question is yes, Does the collection of these theories have a structure?