Let ${\cal M}$ be an o-minimal expansion of a dense linear order $(M,\lt)$.
The first big question about o-minimality is the following: is o-minimality an elementary property, that is, given ${\cal N} \equiv {\cal M}$, is ${\cal N}$ necessarily o-minimal?
Lemma
The following are equivalent:
- every ${\cal N} \equiv {\cal M}$ is o-minimal;
- for every definable $A \subseteq M^{m+n}$, there exists $k \in {\mathbb N}$ such that, for all $x \in M^m$, the fiber $A_x$ is finite if and only if $|A_x| \le k$.
Condition 2, called the uniform finiteness property, is a direct consequence of the cell decomposition theorem proved later. Indeed, special cases of the uniform finiteness property need to be established along with the proof of the cell decomposition theorem.
Proof of the lemma. Assume 1, and let $\phi$ be a formula over $M$ such that $A = \phi\left(M^{m+n}\right)$. By the exercise in this post, we may assume that $A$ has only finite, hence discrete fibers. It follows for any ${\cal N} \succ {\cal M}$ that $\phi\left(N^{m+n}\right)$ has only discrete fibers. So by 1, for any ${\cal N} \succ {\cal M}$ the set $\phi\left(N^{m+n}\right)$ has only finite fibers; part 2 now follows from the compactness theorem.
The converse implication is left as an exercise. $\Box$
Exercise
Assume that ${\cal M}$ is o-minimal and $\aleph_1$-saturated. Show that ${\cal M}$ has the uniform finiteness property.