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A permutation group on a set is oligomorphic when its diagonal action on ordered $n$-tuples has finitely many orbits for every positive integer $n$. For a countable model of a complete theory, an oligomorphic automorphism group makes every finite-arity type space finite: realized types are dense, and only finitely many are possible because types are constant on orbits.

We work with the usual infinite-model convention for an omega-categorical theory: it has a unique countably infinite model up to isomorphism. The three conditions are the type-space and automorphism formulations of the Ryll-Nardzewski theorem; the following arguments prove them rather than invoke that theorem. The standard hypothesis that $T$ has infinite models is understood. If finite models are allowed instead, uniqueness among models of size at most countable is the matching convention for condition (i).

We first prove (i) implies (ii). Suppose some type space $S_n(T)$ were infinite. An infinite compact space cannot have every point isolated: its cover by singleton open sets would have no finite subcover. Thus it has a nonisolated complete type $p$. The omitting types theorem gives a countable model omitting $p$. On the other hand, compactness theorem gives a model realizing $p$, and the Downward Lowenheim-Skolem theorem gives a countable elementary submodel containing its realizing tuple. These countable models cannot be isomorphic, since an isomorphism preserves realization of $p$. This contradicts omega-categoricity. Hence every $S_n(T)$ is finite.

The Stone topology on a type space has basic open setswhere $\phi$ ranges over formulas with parameters in $A$ and $n$ free variables. Here points are complete types consistent with $T_A=\mathrm{Th}(\mathcal{M},a{)}_{a\in A}$. Arbitrary open sets are unions of these basic sets.

The identitiesshow that the basic sets are clopen sets and form a basis. Distinct complete types contain opposite choices for some formula, so this topology is Hausdorff. The compactness theorem gives compactness: a family of basic closed sets with the finite intersection property determines a consistent set of formulas, which extends to a complete type. Thus logical formulas provide a basis of clopen tests for distinguishing points of the type space.

A complete $n$-type over $X\subseteq M$ is a maximal set $p(\bar{x})$ of $L(X)$-formulas consistent with the theory of $M$ with parameters from $X$. Equivalently, for every formula $\phi (\bar{x})$, exactly one of $\phi ,\neg \phi$ belongs to $p$.

The type space $S_n^M(X)$ has these types as points and basic open setsSince $[\phi {]}^c=[\neg \phi ]$, these sets are clopen. An isolated type is a point $\{p\}=[\phi ]$ for some formula $\phi \in p$.

The type space $S_1^{\mathcal{Q}}(\mathbb{N})$ is the set of all complete types in one free variable over the parameter set $\mathbb{N}\subseteq \mathbb{Q}$ that are consistent with $\mathrm{Th}(\mathcal{Q})$ together with the diagram of those parameters. Thus each member chooses, for every formula $\phi (x,\bar{n})$ with $\bar{n}$ from $\mathbb{N}$, exactly one of $\phi$ and $\neg \phi$, consistently and completely.

A Stone space is a compact Hausdorff space that is totally disconnected, equivalently a compact Hausdorff space with a basis of clopen sets. The topology on a type space gives an important example.

For a type space $S_n^M(A)$, the basic open set $[\phi ]$ consists of complete types containing the formula $\phi$. These sets are clopen, satisfy $[\phi ]\cap [\psi ]=[\phi \wedge \psi ]$, and distinguish distinct types. The compactness theorem makes this topology compact; it is Hausdorff and has a basis of clopen sets.

With its standard topology, this type space is a Stone space. The type space $S_n^M(A)$ is the set of complete $n$-types over a parameter set $A$ that are consistent with the complete theory of $M$ together with its diagram over $A$.