When Can Points in Convex Sets be Separated by Affine Maps?

For a class A of convex sets (in not necessarily finite-dimensional) real vector spaces, let Sep A denote the class of all convex sets C such that the affine maps from C to elements of A separate points. If we restrict our attention to finite-dimensional convex sets, there are only four possibilities for SepfA, denoting the intersection of Sep A and {C: C is a finite-dimensional convex set}. Similarly, restriction to absolutely convex sets yields only three possibilities. In the general case, there are many possibilities for Sep A, at least as many as cardinals. In particular, there is no line-free convex set C such that for all linearly bounded convex sets D the affine maps from D to C separate points.