The relation belongs to ;
The elements of are precisely finite unions of points and intervals in .
Here an interval can mean a set of the form , or , or .
A structure on a set can be thought of as the collection of sets that are definable with respect to a one-sorted first-order language with a given interpretation in . Thus is the collection of subsets of which are defined by -ary predicates in . The definition of o-minimal structure supposes that contains a relation symbol , and that is interpreted in as a dense linear order without endpoints.
The o-minimality condition places a sharp restriction on which subsets of can be defined in the language. Essentially, it means that the only definable subsets of are those which are definable in terms of constants and the predicates and .
Remarkably, quite a lot can be said about the structure of definable sets in an o-minimal structure over , and this is a very active area of model theory. The notion of o-minimal structure has been proposed as a reasonable candidate for an axiomatic approach to Grothendieck’s hoped-for “tame topology” ( topologie modérée ).
A theory is o-minimal if every model of is an o-minimal structure.
Michel Coste, An Introduction to O-Minimal Geometry , Lecture notes Pisa 1999. (pdf)
Lou van den Dries, Tame topology and O-minimal structures, London Math. Soc. Lecture Notes Series 248, Cambridge U. Press 1998.
Alexandre Grothendieck, Esquisse d’un Programme, section 5. English translation available in Geometric Galois Actions I (edited by L. Schneps and P. Lochak), LMS Lecture Notes Ser. 242, CUP 1997.
Alessandro Berarducci, Definable groups in o-minimal structures, pdf; Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup, J. Symbolic Logic 74:3 (2009), 891-900, MR2548466, euclid, doi.
Mario J. Edmundo, Luca Prelli, Invariance of o-minimal cohomology with definably compact supports, arxiv/1205.6124