Pro-definability of spaces of definable types

We show pro-definability of spaces definable types in various classical complete first order theories, including o-minimal Presburger arithmetic, $p$-adically closed fields, real and algebraically valued fields ordered differential fields. Furthermore, we prove other distinguished subspaces, some which have an interesting geometric interpretation. Our general strategy consists showing that are uniformly definable, a property implies using argument due to E. Hrushovski F. Loeser. Uniform definability is finally achieved by studying classes stably embedded pairs.