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

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