Given (X,L) a compact Kähler manifold, the study of (degenerate) complex Monge- Ampère equations arises in several questions such as the search of Kähler-Einstein metrics. On the set E1(X, L,T), consisting basically of potentials slightly more singular than T (the prescribed singularities), it is possible to define functionals whose critical points solve the equations.

We will show that E1(X,L,T) has a natural complete metric topology associated to a distance d, and that it can be seen as limit of (E1(X,L,Tk),d) either in a Gromov-Hausdorff sense and in the category of metric spaces when the singularities increase.

Moreover we will prove that, given a family A of nested prescribed singularities, X_A can be equipped of a natural complete distance dA which extends the distances d, and that the Monge-Ampère map MA : X_A -> Y_A becomes a homeomorphism, where Y_A is a certain set of positive Borel measures equipped with a strong topology.

This helps to study the stability of solutions with prescribed singularities of degenerate complex Monge-Ampère equations when the measures and/or the prescribed singularities change. Some examples in the case (X,L) polarized projective manifold, with algebraic singularities will be given and, time permitting, we will see some applications to the (log-) Kähler-Einstein setting.