Definition:Type Space

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Definition

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

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


The space of $n$-types over $A$ is the topological space formed by the set $S_n^{\mathcal M} \left({A}\right)$ together with the topology arising from the basis which consists of the sets:

$[\phi] = \{p \in S_n^{\mathcal M} \left({A}\right):\phi \in p\}$

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 $[\neg\phi]$ in $S_n^{\mathcal M} \left({A}\right)$.


Note

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