Type space of an algebraically closed field over a subfield ($S_n(F)$)

Complete $n$-types over a subfield $F$ of an algebraically closed field correspond to prime ideals of $F[X_1,\dots ,X_n]$, or equivalently to generic points of irreducible affine varieties over $F$. The Stone topology corresponds to the constructible topology: polynomial equalities and inequalities define basic clopen sets.