Let $T$ be a complete first-order theory in a language $\mathcal{L}$, let $\mathfrak{C}$ be a monster model of $T$, let $A \subseteq \mathfrak{C}$ be a parameter set, and let $n \in \mathbb{N}$. Equip the space $S_n(A)$ of complete $n$-types over $A$ with the Stone topology, whose basic open sets are \begin{align*} [\varphi] := \{p \in S_n(A) : \varphi \in p\}, \end{align*} where $\varphi$ ranges over all $\mathcal{L}(A)$-formulas in free variables $x = (x_1,\dots,x_n)$. Then $S_n(A)$ is totally disconnected: every connected subset of $S_n(A)$ has at most one point.

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 $\mathcal{L}$, 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 $p$ be either $0$ or a prime number, and let $\operatorname{ACF}_p$ denote the complete first-order theory of algebraically closed fields of characteristic $p$ in the language of rings. If $K \models \operatorname{ACF}_p$ and $A \subseteq K$ is infinite, then the set $S_1(A)$ of complete $1$-types over $A$ satisfies \begin{align*} |S_1(A)| \leq |A|. \end{align*} Consequently, $\operatorname{ACF}_p$ is $\kappa$-stable for every infinite cardinal $\kappa$ with $\kappa \geq |\mathcal{L}_{\mathrm{ring}}|$.

For a complete theory $T$, the following are equivalent: 1. $T$ is stable in some infinite cardinal $\kappa$ with $\kappa \ge |T|$. 2. For every infinite cardinal $\kappa$ satisfying $\kappa^{|T|} = \kappa$, the theory $T$ is stable in $\kappa$. 3. No formula has the order property with respect to $T$.

Let $L$ be a first-order language, and let $T$ be a complete $L$-theory with quantifier elimination. Let $M,N \models T$. Let $A$ be a parameter set equipped with interpretations $i_M:A \to M$ and $i_N:A \to N$ whose named constants have matching quantifier-free diagram in $M$ and $N$; for example, $A$ may be a common substructure of both models with the same interpretation of all $L$-terms. If tuples $b \in M^n$ and $c \in N^n$ have the same quantifier-free type over $A$ via the interpretations $i_M$ and $i_N$, then they have the same complete type over $A$ via those interpretations.

Let $L$ be a countable first-order language, and let $T$ be a complete $L$-theory with infinite models. For each $n \geq 1$, let $S_n(T)$ denote the Stone space of complete $n$-types over $\varnothing$ consistent with $T$. The following conditions are equivalent: 1. $T$ is $\omega$-categorical, meaning that any two countable models of $T$ are isomorphic. 2. For every $n \geq 1$, the Stone space $S_n(T)$ is finite. 3. For every $n \geq 1$, there are only finitely many $L$-formulas with free variables among $x_1,\dots,x_n$, up to equivalence modulo $T$. 4. For some countable model $M \models T$, the automorphism group $\operatorname{Aut}(M)$ is oligomorphic, meaning that for every $n \geq 1$, the componentwise action of $\operatorname{Aut}(M)$ on $M^n$ has only finitely many orbits.

Let $L$ be a countable first-order language, let $T$ be a consistent $L$-theory, and for each $i \in \mathbb{N}$ let $p_i(x_i)$ be a partial type over the empty set, where $x_i$ is a finite tuple of variables. Suppose that no $p_i$ is principal over $T$: that is, for every $L$-formula $\chi(x_i)$ such that $T \cup \{\exists x_i\,\chi(x_i)\}$ is consistent, there is some $\varphi(x_i) \in p_i$ such that \begin{align*} T \not\models \forall x_i\,(\chi(x_i) \to \varphi(x_i)). \end{align*} Then there exists a countable $L$-structure $M$ such that $M \models T$ and, for every $i \in \mathbb{N}$, no tuple $a \in M^{|x_i|}$ realizes $p_i$; equivalently, for every $i \in \mathbb{N}$ and every $a \in M^{|x_i|}$, there is some $\varphi(x_i) \in p_i$ such that $M \models \neg \varphi(a)$.

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 $F$ 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 $\kappa \geq |F| + \aleph_0$.