I am searching for the "most natural" definition of a (geometrical/space) point as an element of "something" in mathematics (I am trying to design a small computational geometry library on strong mathematical basis). For example, for a vector, it's easy: a vector is an element of a vector space, end of story. Same for a tensor: a tensor is an element of the tensor product of vector spaces. But how to define a point in the same way?
Can a point be defined as an element of an affine space or as an element of a topological space?
If both are true, what are the difference between the two types of points, and what would be the most natural (it's subjective) approach to define a point in geometry?
Moreover, do other approaches exist (a point is an element of XXXX)?
EDIT: I know that a point is an axiom/primitive notion but to design the library I need to make a choice. And I want to know the best option...