A tameness condition for ordered first-order structures requiring that every definable subset of the line (one-dimensional domain) is a finite union of points and open intervals. It formalizes a lack of pathological oscillation in definable sets of the order.

Principle

Restrict one-dimensional definable sets to simple topological pieces (points and intervals), which forces regularity in geometric and combinatorial behavior and enables cell decomposition and dimension theory.

Demonstration

Concrete example: the ordered field of real numbers with its field operations (a real closed field) is o-minimal when considered with the semialgebraic language; definable subsets are finite unions of points and intervals, and higher-dimensional definable sets admit cell decomposition into manifold-like cells.

Misapplication

Treating an expansion by arbitrary functions (for example adding unrestricted analytic oscillatory functions) as still o-minimal without verifying definability conditions; assuming o-minimality in multi-sorted or non-ordered contexts where the one-dimensional criterion does not apply.

Consequence

O-minimality yields strong geometric and topological regularity: dimension theory, finiteness of definably connected components, tame measure and triangulation properties, and often a well-behaved notion of definable continuity and differentiability.

Reversal

The opposite notion allows arbitrary definable subsets of the line, including dense and highly oscillatory sets with fractal-like behavior and infinite alternation of points and intervals, destroying cell decomposition.

Boundary

A property of ordered structures and their first-order expansions; it is explicitly about one-dimensional definable sets and does not automatically guarantee tameness for all sorts or for expansions that add higher complexity functions or predicates.

Semantic Tension

Tension arises with weaker tameness notions (weak o-minimality, quasi-o-minimality) and with analytic expansions: some expansions preserve o-minimality while others increase definable complexity dramatically, making fine distinctions necessary.

Synthesis

O-minimality is a one-dimensional tameness axiom for ordered structures forcing definable sets to decompose into finitely many intervals and points, which in turn supports a rich geometric theory (cell decomposition, dimension, regularity) for definable sets and maps.