o-minimality

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:

• for some $a,b$ from $M$

• $\{x:x>a\}$ for some $a$ from $M$

• 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 $M\equiv N$ 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 $M^n$ for $n\in \omega$.