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.