structural:scholar_edition, linguistic:voice_change, linguistic:paraphrasing
Stone spaces

The set of definable subsets of M n {\displaystyle {\mathcal {M}}^{n}} over some parameters A {\displaystyle A} is a Boolean algebra. By Stone's representation theorem for Boolean algebras there is a natural dual topological space, which consists exactly of the complete n {\displaystyle n} -types over A {\displaystyle A}. The topology generated by sets of the form { p | φ ∈ p } {\displaystyle \{p|\varphi \in p\}} for single formulas φ {\displaystyle \varphi }. This is called the Stone space of n-types over A. This topology explains some of the terminology used in model theory: The compactness theorem says that the Stone space is a compact topological space, and a type p is isolated if and only if p is an isolated point in the Stone topology.

While types in algebraically closed fields correspond to the spectrum of the polynomial ring, the topology on the type space is the constructible topology: a set of types is basic open iff it is of the form { p: f (x) = 0 ∈ p } {\displaystyle \{p:f(x)=0\in p\}} or of the form { p: f (x) ≠ 0 ∈ p } {\displaystyle \{p:f(x)\neq 0\in p\}}. This is finer than the Zariski topology.