# Definition:Isolated Type

## Definition

Let $T$ be an $\mathcal L$-theory.

Let $\phi \left({\bar v}\right)$ be an $\mathcal L$-formula in $n$ free variables $\bar v$ such that $T \cup \phi \left({\bar v}\right)$ is satisfiable.

Let $p$ be an $n$-type of $T$.

We say that $\phi$ isolates $p$ if for all $\psi \in p$, we have:

$T \models \forall \bar{v} \left({ \phi \left({\bar v}\right) \rightarrow \psi \left({\bar v}\right) }\right)$

that is, all $\psi$ are semantic consequences of $\phi$.