Let $F$ be a field, let $\mathcal{L}_F$ be the language of $F$-modules, and let $T$ be the complete $\mathcal{L}_F$-theory of infinite-dimensional vector spaces over $F$. If $V \models T$ and $A \subset V$ is an infinite parameter set, then the set $S_1^T(A)$ of complete $1$-types over $A$ satisfies \begin{align*} |S_1^T(A)| \leq |A| + |F|. \end{align*} Consequently, $T$ is $\kappa$-stable for every infinite cardinal $\kappa$ such that $\kappa \geq |F| + \aleph_0$.

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 strongly minimal theory and let $M \models T$. Then the [algebraic closure](/page/Algebraic%20Closure) operator $A \mapsto \operatorname{acl}_M(A)$ defines a pregeometry on the underlying set of $M$.

Let $T$ be a complete theory in a language $L$, let $A$ be a set of parameters from a model of $T$, and let $L(A)$ be the language obtained from $L$ by adding constant symbols for elements of $A$. Write \begin{align*} T_A := T \cup \operatorname{Diag}(A). \end{align*} Let $p(x)$ be a complete $n$-type over $A$, where $x = (x_1,\dots,x_n)$, and let $\theta(x)$ be an $L(A)$-formula. Then $\theta(x)$ isolates $p(x)$ if and only if $\theta(x) \in p(x)$ and, for every $L(A)$-formula $\varphi(x)$, \begin{align*} \varphi(x) \in p(x) \quad \Longleftrightarrow \quad T_A \models \forall x\,(\theta(x) \to \varphi(x)). \end{align*} Equivalently, $\theta(x) \in p(x)$ and $\theta(x)$ decides every $L(A)$-formula over $T_A$, with the chosen decision agreeing with membership in $p(x)$.

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 strongly minimal and let $M\models T$. For every $A\subseteq M$, every set $I\subseteq M$ that is independent over $A$ extends to a basis of $M$ over $A$, and any two bases of $M$ over $A$ have the same cardinality.

Let $T$ be a complete first-order theory in a language $\mathcal{L}$, let $M \models T$, let $A \subset M$, and let $n \in \mathbb{N}$. Equip the set $S_n(A)$ of complete $n$-types over $A$ with the topology generated by the basic sets \begin{align*} [\varphi] := \{p \in S_n(A) : \varphi(x_1,\dots,x_n) \in p\}, \end{align*} where $\varphi(x_1,\dots,x_n)$ ranges over all $\mathcal{L}(A)$-formulas with free variables among $x_1,\dots,x_n$. Then $S_n(A)$ is compact, Hausdorff, and totally disconnected. In fact, the sets $[\varphi]$ form a basis of clopen subsets.

Let $T$ be a complete first-order theory in a countable language. Then $T$ is totally transcendental if and only if $T$ is $\omega$-stable. Moreover, \begin{align*} \omega\text{-stable} \implies \text{superstable} \implies \text{stable}. \end{align*} Both implications are strict among complete countable first-order theories: there exists a complete countable superstable theory that is not $\omega$-stable, and there exists a complete countable stable theory that is not superstable.

Let $T$ be a complete first-order theory in a countable language. Suppose there exists an uncountable cardinal $\kappa$ such that any two models of $T$ of cardinality $\kappa$ are isomorphic. Then $T$ is stable; equivalently, no formula of $T$ has the order property.