Definition:Type Space

Definition
Let $\mathcal{M}$ be an $\mathcal{L}$-structure, and let $A$ be a subset of the universe of $\mathcal{M}$.

Let $S_{n}^{\mathcal{M}}(A)$ be the set of complete $n$-types over A.

The space of $n$-types over $A$ is the topological space formed by the set $S_{n}^{\mathcal{M}}(A)$ together with the topology arising from the basis which consists of the sets $[\phi] = \{p \in S_{n}^{\mathcal{M}}(A):\phi \in p\}$ for each $\mathcal{L}_A$-formula $\phi$ with $n$ free variables.

Note that each $[\phi]$ is also closed in this topology, since $[\phi]$ is the complement of $[\neg\phi]$.

Note
This is also referred to as the Stone space of $S_{n}^{\mathcal{M}}(A)$, since it is an example of this more general construction for Boolean algebras.