In model theory, the concept of a saturated model can be thought of as a process of adjoining `limit points'. For example a saturated model of $\mathbb Q$ with coefficients in $\mathbb Q$ is a certain linear order extending $\mathbb R$; containing all the dedekind cuts which are classically thought of as filling in the 'holes' in $\mathbb Q$.
Is there a worked out reasonable context, in which we can think of projective space $\mathbb P^k_n$ as a "saturated" model of affine space $\mathbb A^n_k$, in the sense of adding in the points of infinity as limit points?
I hope that such a model would capture $\mathbb P^k_n$ as a algebro-geometric object, perhaps capitalizing on the fact that over an algebraically closed field $k$, the type space $S_n(L)$ for a field $L\subset k$ is topologically the same as $\mathbb A^n_L$ except that the principal closed sets $V(I)$ for an ideal $I\subset L[x_1,...,x_n]$ are both open as well as closed.