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

For a class $\mathcal{A}$ of convex sets (in not necessarily finite-dimensional) real vector spaces, let Sep $\mathcal{A}$ denote the class of all convex sets C such that the affine maps from C to elements of $\mathcal{A}$ separate points. If we restrict our attention to finite-dimensional convex sets, there are only four possibilities for $\mathrm{Sep}_\mathrm{f}\mathcal{A} := \mathrm{Sep}\,\mathcal{A} \cap \{\mathcal{C} : \mathcal{C} \text{ finite-dimensional convex set}\}$. Similarly, restriction to absolutely convex sets yields only three possibilities. In the general case, there are many possibilities for Sep $\mathcal{A}$, 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.