O-minimal theory

In mathematical logic, and more specifically in model theory, an infinite structure ("M",<...) which is totally ordered by < is called an o-minimal structure if and only if every definable subset "X" ⊂ "M" (with parameters taken from "M") is a finite union of intervals and points.
O-minimality can be regarded as a weak form of quantifier elimination. A structure "M" is o-minimal if and only if every formula with one free variable and parameters in "M" is equivalent to a quantifier-free formula involving only the ordering, also with parameters in "M". This is analogous to the minimal structures, which are exactly the analogous property down to equality.
A theory "T" is an o-minimal theory if every model of "T" is o-minimal. It is known that the complete theory "T" of an o-minimal structure is an o-minimal theory. This result is remarkable because, in contrast, the complete theory of a minimal structure need not be a strongly minimal theory, that is, there may be an elementarily equivalent structure which is not minimal.
Set-theoretic definition.
O-minimal structures can be defined without recourse to model theory. Here we define a structure on a nonempty set "M" in a set-theoretic manner, as a sequence "S" = ("S""n"), "n" = 0,1,2... such that
If "M" has a dense linear order without endpoints on it, say <, then a structure "S" on "M" is called o-minimal if it satisfies the extra axioms
The "o" stands for "order", since any o-minimal structure requires an ordering on the underlying set.
Model theoretic definition.
O-minimal structures originated in model theory and so have a simpler — but equivalent — definition using the language of model theory. Specifically if "L" is a language including a binary relation <, and ("M",<...) is an "L"-structure where < is interpreted to satisfy the axioms of a dense linear order, then ("M",<...) is called an o-minimal structure if for any definable set "X" ⊆ "M" there are finitely many open intervals "I"1..., "I""r" without endpoints in "M" ∪ {±∞} and a finite set "X"0 such that
Examples.
Examples of o-minimal theories are:
In the case of RCF, the definable sets are the semialgebraic sets. Thus the study of o-minimal structures and theories generalises real algebraic geometry. A major line of current research is based on discovering expansions of the real ordered field that are o-minimal. Despite the generality of application, one can show a great deal about the geometry of set definable in o-minimal structures. There is a cell decomposition theorem, Whitney and Verdier stratification theorems and a good notion of dimension and Euler characteristic.