[proofplan] We separate two distinct complete types by finding a formula on which they disagree. Completeness then forces one type to contain the formula and the other to contain its negation. The corresponding basic clopen sets in the Stone topology are disjoint open neighbourhoods, giving the Hausdorff property. [/proofplan] [step:Choose a formula that separates the two distinct complete types] Let $p,q \in S_n(A)$ satisfy $p \neq q$. Since $p$ and $q$ are distinct sets of $L(A)$-formulas in variables $x_1,\dots,x_n$, there exists an $L(A)$-formula $\varphi(x_1,\dots,x_n)$ such that, after possibly interchanging $p$ and $q$, \begin{align*} \varphi \in p \qquad\text{and}\qquad \varphi \notin q. \end{align*} Because $q$ is a complete type over $A$, exactly one of $\varphi$ and $\neg \varphi$ belongs to $q$. Since $\varphi \notin q$, we have \begin{align*} \neg \varphi \in q. \end{align*} [guided] Let $p,q \in S_n(A)$ be distinct points of the type space. A point of $S_n(A)$ is a complete type over $A$, hence a complete consistent set of $L(A)$-formulas in the variables $x_1,\dots,x_n$. The inequality $p \neq q$ therefore means that these two sets of formulas are not equal. Consequently, there is an $L(A)$-formula $\varphi(x_1,\dots,x_n)$ that belongs to one of the two types but not the other. Interchanging the names of $p$ and $q$ if necessary, we may assume \begin{align*} \varphi \in p \qquad\text{and}\qquad \varphi \notin q. \end{align*} Now we use completeness of the type $q$. Completeness means that for every $L(A)$-formula $\psi(x_1,\dots,x_n)$, exactly one of $\psi$ and $\neg \psi$ belongs to $q$. Applying this with $\psi=\varphi$, and using $\varphi \notin q$, we obtain \begin{align*} \neg \varphi \in q. \end{align*} This is the model-theoretic reason the Stone topology separates distinct complete types: disagreement on one formula becomes membership in complementary basic opens. [/guided] [/step] [step:Use complementary basic clopen sets as disjoint neighbourhoods] Define the two basic Stone-open sets \begin{align*} [\varphi] &:= \{r \in S_n(A) : \varphi \in r\},\\ [\neg\varphi] &:= \{r \in S_n(A) : \neg\varphi \in r\}. \end{align*} By the definition of the Stone topology, both $[\varphi]$ and $[\neg\varphi]$ are open subsets of $S_n(A)$. From $\varphi \in p$ and $\neg\varphi \in q$, we have \begin{align*} p \in [\varphi] \qquad\text{and}\qquad q \in [\neg\varphi]. \end{align*} The sets $[\varphi]$ and $[\neg\varphi]$ are disjoint: if $r \in [\varphi] \cap [\neg\varphi]$, then the complete type $r$ contains both $\varphi$ and $\neg\varphi$, contradicting consistency of $r$. Therefore $p$ and $q$ have disjoint open neighbourhoods. Since $p$ and $q$ were arbitrary distinct points of $S_n(A)$, the space $S_n(A)$ is Hausdorff. [/step]