Elementary extension

An extension $N\supseteq M$ is elementary when every formula with parameters from $M$ has the same truth value in both structures.

The complete type of a tuple over parameters records every formula over those parameters that the tuple satisfies.

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 $A$.

A complete type $p\in S_n(A)$ is isolated when some formula $\phi \in p$ belongs to no other complete type in $S_n(A)$. Equivalently, the basic open set determined by $\phi$ is the singleton $\{p\}$.

Quantifier elimination for dense linear orders shows that the one-types over $\mathbb{N}\subset \mathbb{Q}$ are determined by cuts: equality to a natural number, one of the intervals between consecutive natural numbers, the ray below zero, or the cut above every natural number. Only the last type is non-isolated.

A model is aleph-zero-homogeneous when every finite partial elementary map extends by one more element.