If $p\neq q$, completeness gives a formula $\varphi$ with $\varphi\in p$ and $\neg\varphi\in q$. Then
$$
p\in[\varphi],\qquad q\in[\neg\varphi],
$$
and these two disjoint open sets cover $S_n^M(X)$. This proves the <total disconnectedness of a type space>.

Solution

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

Distinct complete types contain opposite choices $\varphi$ and $\neg\varphi$. The corresponding basic clopen sets partition the type space, so every type space is totally disconnected.


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