Let $M$ be a prime model and let $[\varphi]\subseteq S_n(T)$ be a nonempty basic open set. Some model of $T$ realizes $\varphi$, so completeness of $T$ gives
$$
T\models\exists\bar x\,\varphi(\bar x).
$$
Therefore $M$ realizes $\varphi$ by some tuple $\bar a$. The prime-model characterization in part (e) says $\operatorname{tp}^M(\bar a/\varnothing)$ is isolated, and it lies in $[\varphi]$. Every nonempty basic open set thus contains an isolated point, proving <density of isolated types from a prime model>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 120 / 2 / f / Solution

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.


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