Let $T$ be a complete countable stable theory, let $A$ be a small parameter set, and let $D$ be an $A$-definable strongly minimal set. Assume that for every independent set $I \subset D$ there is a model $M_I$ prime and atomic over $A \cup I$, and that finite independent tuples from $D$ of the same length have the same type over $A$. If $I,J \subset D$ are independent sets with $|I|=|J|$, then $M_I \cong_A M_J$.

Let $T$ be a complete theory in a language $L$, let $M\models T$, and let $\kappa$ be an infinite cardinal such that $|L|<\kappa$, $\kappa^{<\kappa}=\kappa$, and $|M|\le \kappa$. Then there is an elementary extension $N\succeq M$ such that $|N|\le \kappa$ and $N$ realizes every complete type in finitely many variables over every parameter set $A \subset N$ with $|A| < \kappa$.

Let $T$ be a complete first-order theory in a language $\mathcal{L}$. If $M \models T$ is countable and atomic, meaning that for every finite tuple $a \in M^n$ the complete type $\operatorname{tp}^M(a/\varnothing)$ is isolated by an $\mathcal{L}$-formula, then $M$ is a prime model of $T$: for every model $N \models T$, there exists an elementary embedding $f: M \to N$.

Let $L$ be a first-order language, let $M$ be an $L$-structure, and let $\kappa$ be an infinite cardinal such that $|L|<\kappa$, $\kappa^{<\kappa}=\kappa$, and $|M|\le \kappa$. Then there is an elementary extension $N\succeq M$ such that $|N|\le \kappa$ and $N$ is $\kappa$-saturated.

Let $T$ be a complete first-order theory in a countable language $L$. Suppose that for every integer $n \geq 1$ and every non-isolated complete type $p(x_1,\dots,x_n) \in S_n(T)$, there exists a countable model $M \models T$ such that no tuple from $M^n$ realizes $p$. If $T$ has exactly one countable model up to isomorphism, then for every integer $n \geq 1$, every complete type $p(x_1,\dots,x_n) \in S_n(T)$ that is realized in some model of $T$ is isolated.

Let $L$ be a first-order language, let $A$ be a set of parameters for $L$, and let $n \in \mathbb{N}$. For each $L(A)$-formula $\varphi(\bar{x})$ in the tuple of variables $\bar{x} = (x_1,\dots,x_n)$, define \begin{align*} [\varphi] := \{p \in S_n(A) : \varphi(\bar{x}) \in p\}. \end{align*} Then the family \begin{align*} \mathcal{B} := \{[\varphi] : \varphi(\bar{x}) \text{ is an } L(A)\text{-formula}\} \end{align*} is a basis for the Stone topology on $S_n(A)$, and every member of $\mathcal{B}$ is clopen.

Let $T$ be a complete first-order theory in a language $L$, let $\mathfrak{C} \models T$ be a sufficiently saturated monster model, let $A \subseteq \mathfrak{C}$ be a parameter set, and let $n \in \mathbb{N}$. Let $S_n(A)$ denote the set of complete $n$-types over $A$, equipped with the Stone topology whose basic clopen subsets are \begin{align*} [\varphi(x)] := \{p \in S_n(A) : \varphi(x) \in p\}, \end{align*} where $x = (x_1,\dots,x_n)$ and $\varphi(x)$ ranges over all $L(A)$-formulas in the variables $x$. Then $S_n(A)$ is compact.

Let $T$ be a complete theory in a countable language $\mathcal{L}$. For each integer $n \geq 1$, let $S_n(T)$ denote the Stone space of complete $n$-types over $T$, with basic open sets of the form \begin{align*} [\varphi] := \{p \in S_n(T) : \varphi \in p\} \end{align*} for $\mathcal{L}$-formulas $\varphi(x_1,\dots,x_n)$. Then $T$ has a countable atomic model if and only if, for every integer $n \geq 1$, the isolated types are dense in $S_n(T)$.

Let $T$ be stable, let $\mathfrak C \models T$ be a monster model, let $M \preceq \mathfrak C$ be $\kappa$-saturated, and let $B=M\cup A\subseteq\mathfrak C$ with $|A|<\kappa$. Every type $p(x)\in S_x(M)$ has an extension $q(x)\in S_x(B)$ that is the restriction of a global $M$-invariant type built from an $M$-definitional scheme for $p$.

Let $T$ be a complete countable theory. If $T$ is categorical in an uncountable cardinal $\kappa$, then its unique model of cardinality $\kappa$ is $\kappa$-saturated.