O-minimal theory

Remarks

Example: the ordered field $(\R, +, \cdot, <, 0, 1)$ is o-minimal. By the Tarski–Seidenberg theorem it has quantifier elimination, so a definable subset of $\R$ is a Boolean combination of sets $\set{x : p(x) > 0}$ and $\set{x : p(x) = 0}$ with $p$ a polynomial with real coefficients. Each such set is a finite union of points and open intervals, because a nonzero polynomial has finitely many roots.

Non-example: the structure $(\Z, <, +)$ is not o-minimal. The formula $\exists y \, (x = y + y)$ defines the set of even integers, which is not a finite union of points and intervals.

By a theorem of Knight, Pillay and Steinhorn, if a structure $M$ is o-minimal then its complete theory is o-minimal. Dense linear orders without endpoints and real closed fields give o-minimal theories.