Let M be an ordered structureMathworldPlanetmath. An interval in M is any subset of M that can be expressed in one of the following forms:

  • {x:a<x<b} for some a,b from M

  • {x:x>a} for some a from M

  • {x:x<a} for some a from M

Then we define M to be o-minimal iff every definable subset of M is a finite union of intervals and points. This is a property of the theory of M i.e. if MN and M is o-minimal, then N is o-minimal. Note that M being o-minimal is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to every definable subset of M being quantifier free definable in the languagePlanetmathPlanetmath with just the ordering. Compare this with strong minimality.

The model theoryMathworldPlanetmath of o-minimal structures is well understood, for an excellent account see Lou van den Dries, Tame topology and o-minimal structures, CUP 1998. In particular, although this condition is merely on definable subsets of M it gives very good information about definable subsets of Mn for nω.

Title o-minimality
Canonical name Ominimality
Date of creation 2013-03-22 13:23:01
Last modified on 2013-03-22 13:23:01
Owner Timmy (1414)
Last modified by Timmy (1414)
Numerical id 7
Author Timmy (1414)
Entry type Definition
Classification msc 03C64
Classification msc 14P10
Related topic StronglyMinimal
Defines o-minimal