Let $T$ be a complete strongly minimal first-order theory in a language $\mathcal{L}$, and let $M,N \models T$. Write \begin{align*} A_M &:= \operatorname{acl}^M(\varnothing),& A_N &:= \operatorname{acl}^N(\varnothing). \end{align*} Let $B \subseteq M$ be a basis of $M$ over $A_M$ for the pregeometry induced by [algebraic closure](/page/Algebraic%20Closure), and let $C \subseteq N$ be a basis of $N$ over $A_N$. Suppose that $g:A_M \to A_N$ is an elementary bijection and that $f:B \to C$ is a bijection. Then the map $g \cup f:A_M \cup B \to A_N \cup C$ extends to an $\mathcal{L}$-isomorphism \begin{align*} M \cong N. \end{align*} Consequently, after fixing the elementary bijection $g:A_M \to A_N$, the models $M$ and $N$ are isomorphic by an isomorphism extending $g$ if and only if $\dim(M)=\dim(N)$…

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 $L$, let $\mathfrak{C}$ be a monster model of $T$, let $A \subseteq \mathfrak{C}$ be a parameter set, and let $n \in \mathbb{N}$. Then the type space $S_n(A)$ of complete $n$-types over $A$, equipped with the Stone topology whose basic clopen sets are \begin{align*} [\varphi] := \{p \in S_n(A) : \varphi \in p\}, \end{align*} for $L(A)$-formulas $\varphi(x_1,\dots,x_n)$, is Hausdorff.

Let $T$ be a complete countable first-order theory in a countable language $L$, and suppose that $T$ is totally transcendental. Let $\mathfrak{C} \models T$ be a sufficiently saturated monster model, and let $A \subseteq \mathfrak{C}$ be countable. Let $L(A)$ be the expansion of $L$ by a constant symbol $c_a$ for each $a \in A$, and let $T_A$ be the complete $L(A)$-theory of $\mathfrak{C}$ with $c_a$ interpreted as $a$. Then $T_A$ has a prime model. Equivalently, there exists an elementary substructure $M \preccurlyeq \mathfrak{C}$ with $A \subseteq M$ such that for every model $N \models T$ containing $A$, there is an elementary embedding $f: M \to N$ satisfying $f(a) = a$ for every $a \in A$. Moreover, any two models prime over $A$ are isomorphic over $A$.

Let $A$ be a parameter set, let $x$ be a tuple of variables, and let $S_x(A)$ denote the Stone space of complete $x$-types over $A$, equipped with the Stone topology whose basic clopen sets are \begin{align*} [\psi] := \{p \in S_x(A) : \psi \in p\} \end{align*} for $A$-formulas $\psi(x)$. For every $A$-formula $\varphi(x)$, define \begin{align*} f_\varphi: S_x(A) &\to \{0,1\} \\ p &\mapsto \begin{cases} 1, & \varphi \in p, \\ 0, & \varphi \notin p. \end{cases} \end{align*} where $\{0,1\}$ is equipped with the discrete topology. Then $f_\varphi$ is continuous.

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 $T$ 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 $D$ 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.

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 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*}