stylistic:tone
Stone spaces

It is useful to consider the set of complete n -types over A as a topological space. Consider the following equivalence relation on formulas in the free variables x,..., x with parameters in A:

One can show that ψ ≡ ϕ {\displaystyle \psi \equiv \phi } if and only if they are contained in exactly the same complete types.

The set of formulas in free variables x,..., x over A up to this equivalence relation is a Boolean algebra (and is canonically isomorphic to the set of A -definable subsets of M). The complete n -types correspond to ultrafilters of this Boolean algebra. The set of complete n -types can be made into a topological space by taking the sets of types containing a given formula as a basis of open sets. This constructs the Stone space associated to the Boolean algebra, which is a compact, Hausdorff, and totally disconnected space.

Example. The complete theory of algebraically closed fields of characteristic 0 has quantifier elimination, which allows one to show that the possible complete 1-types (over the empty set) correspond to:

In other words, the 1-types correspond exactly to the prime ideals of the polynomial ring Q over the rationals Q: if r is an element of the model of type p, then the ideal corresponding to p is the set of polynomials with r as a root (which is only the zero polynomial if r is transcendental). More generally, the complete n -types correspond to the prime ideals of the polynomial ring Q, in other words to the points of the prime spectrum of this ring. (The Stone space topology can in fact be viewed as the Zariski topology of a Boolean ring induced in a natural way from the Boolean algebra. While the Zariski topology is not in general Hausdorff, it is in the case of Boolean rings.) For example, if q (x, y) is an irreducible polynomial in two variables, there is a 2-type whose realizations are (informally) pairs (x, y) of elements with q (x, y)=0.