---
tag: DTWHAW1N0V
kind: concept
name: O-minimal theory
msc: [03C64]
sources:
  - url: https://en.wikipedia.org/wiki/O-minimal_theory
same_as:
  wikidata: Q7072039
  wikipedia: O-minimal theory
---

## Summary

An o-minimal structure is a [[totally ordered|0FRXBVVR03]] structure in which every definable subset of the line is a finite union of points and open [[intervals|0Q1D85B8G6]].
An o-minimal theory is a theory all of whose models are o-minimal.
This condition on sets in one variable forces tame behaviour in all dimensions.
Definable sets admit cell decompositions and have a good dimension theory.
The ordered field of real numbers is the model case.
O-minimality is the basis of "tame geometry", used in real algebraic geometry and in number theory.

## Explanations

### L1 human

In an o-minimal structure, a formula about one number can only describe a set made of finitely many points and intervals.
You can never describe an infinite discrete set such as the integers, or a set with infinitely many gaps.
For the real numbers with addition, multiplication and order, the inequality $x^2 - 2 > 0$ describes $(-\infty, -\sqrt 2) \cup (\sqrt 2, \infty)$, a union of two intervals.
Adding the sine function would break this, because the zeros of sine form an infinite discrete set.