Classifying singularities up to analytic extensions of scalars is smooth

The singularity space consists of all germs (X,x), with X a Noetherian scheme and x a point, where we identify two such germs if they become the same after an analytic extension of scalars. This is a complete, separable space for the metric given by the order to which jets (=infinitesimal neighborhoods) agree after base change. In the terminology of descriptive set-theory, the classification of singularities up to analytic extensions of scalars is a smooth problem. Over C, the following two classification problems up to isomorphism are then also smooth: (i) analytic germs; and (ii) polarized schemes.