Definition:Type Space

Definition

Let $\MM$ be an $\LL$-structure, and let $A$ be a subset of the universe of $\MM$.

Let $\map {S_n^\MM} A$ be the set of complete $n$-types over $A$.

The space of $n$-types over $A$ is the topological space formed by the set $\map {S_n^\MM} A$ together with the topology arising from the basis which consists of the sets:

$\sqbrk \phi := \set {p \in \map {S_n^\MM} A:\phi \in p}$

for each $\LL_A$-formula $\phi$ with $n$ free variables.

Note that each $\sqbrk \phi$ is also closed in this topology, since $\sqbrk \phi$ is the complement of $\sqbrk {\neg \phi}$ in $\map {S_n^\MM} A$.

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Also known as

This is also referred to as the Stone space of $\map {S_n^\MM} A$, since it is an example of this more general construction for Boolean algebras.