Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 23 / 5 / b

The Stone topology on a type space has basic open setswhere $\phi$ ranges over formulas with parameters in $A$ and $n$ free variables. Here points are complete types consistent with $T_A=\mathrm{Th}(\mathcal{M},a{)}_{a\in A}$. Arbitrary open sets are unions of these basic sets.

The identitiesshow that the basic sets are clopen sets and form a basis. Distinct complete types contain opposite choices for some formula, so this topology is Hausdorff. The compactness theorem gives compactness: a family of basic closed sets with the finite intersection property determines a consistent set of formulas, which extends to a complete type. Thus logical formulas provide a basis of clopen tests for distinguishing points of the type space.