Let M be an ordered structure
. 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 M≡N and M is o-minimal, then N is o-minimal. Note that M being o-minimal is equivalent to every definable subset of M being quantifier free definable in the language with just the ordering. Compare this with strong minimality.
The model theory 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 n∈ω.