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 $F$ 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 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 $A$ 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 first-order theory in a language $L$, and work inside a sufficiently saturated monster model $\mathfrak{C} \models T$. Suppose that for every parameter set $A \subseteq \mathfrak{C}$ with $A$ infinite and \begin{align*} |A| \geq |L| + \aleph_0, \end{align*} there exist a set $D_A$ and an injective map \begin{align*} \Phi_A: S_1(A) &\to D_A \end{align*} such that \begin{align*} |D_A| \leq |A|. \end{align*} Then, for every infinite cardinal $\kappa$ with $\kappa \geq |L| + \aleph_0$, the theory $T$ is $\kappa$-stable, in the sense that for every parameter set $A \subseteq \mathfrak{C}$ with $|A| = \kappa$, \begin{align*} |S_1(A)| \leq \kappa. \end{align*}

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 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 that there exists an uncountable cardinal $\kappa$ such that $T$ is $\kappa$-categorical, meaning that any two models of $T$ of cardinality $\kappa$ are isomorphic. Then no formula $\varphi(x;y)$ has the order property modulo $T$: there do not exist a model $M \models T$ and sequences $(a_i)_{i \in \mathbb{N}}$ and $(b_j)_{j \in \mathbb{N}}$ of finite tuples from $M$, with $|a_i| = |x|$ and $|b_j| = |y|$, such that for all $i,j \in \mathbb{N}$, \begin{align*} M \models \varphi(a_i;b_j) \quad \iff \quad i < j. \end{align*}

Let $D$ be a parameter-free definable strongly minimal set in a monster model $\mathfrak{M}$ of a complete theory. For $A \subset D$, define \begin{align*} \operatorname{cl}: \mathcal{P}(D) \to \mathcal{P}(D), \qquad \operatorname{cl}(A)=D \cap \operatorname{acl}_{\mathfrak{M}}(A). \end{align*} Then $\operatorname{cl}$ is a pregeometry on $D$: it is extensive, increasing, idempotent, has finite character, and satisfies exchange.