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

A complete $n$-type over $X\subseteq M$ is a maximal set $p(\bar{x})$ of $L(X)$-formulas consistent with the theory of $M$ with parameters from $X$. Equivalently, for every formula $\phi (\bar{x})$, exactly one of $\phi ,\neg \phi$ belongs to $p$.

The type space $S_n^M(X)$ has these types as points and basic open setsSince $[\phi {]}^c=[\neg \phi ]$, these sets are clopen. An isolated type is a point $\{p\}=[\phi ]$ for some formula $\phi \in p$.