Let T be a complete L-theory and M be a model of T. Let x be a tuple of variables and S_x(M) be the space of types over M with free variables x. In this talk we will be interested in the subset S_x^def(M) of S_x(M) of definable types. We will show that for various classical first order theories, including o-minimal expansions of divisible abelian groups, Presburger arithmetic, p-adically closed fields, real closed and algebraically closed valued fields and closed ordered differential fields, the space S_x^def(M)$ is pro-definable, i.e., a projective limit of definable sets.
Our general strategy consists in studying the class of stably embedded pairs of models of the T. Pro-definability is obtained by showing that such class is elementary in the language of pairs.
This is joint work with Jinhe Ye. | 
