Isolated type

A complete type $p\in S_n(A)$ is isolated when some formula $\phi \in p$ belongs to no other complete type in $S_n(A)$. Equivalently, the basic open set determined by $\phi$ is the singleton $\{p\}$.

A principal complete type is isolated by one first-order consistent first-order formula: modulo the first-order theory, that logical formula implies every member of the partial type. For complete types this is exactly an isolated type. The qualifier complete distinguishes this notion from partial types.

If a complete countable theory has a prime model, isolated types are dense in every finite-arity type space. Every nonempty basic open set is realized in the prime model, whose finite tuples have isolated types.