Weakly o-minimal structure

In model theory, a weakly o-minimal structure is a model theoretic structure whose definable sets in the domain are just finite unions of convex sets.
Definition.
A linearly ordered structure, "M", with language "L" including an ordering relation <, is called weakly o-minimal if every parametrically definable subset of "M" is a finite union of convex (definable) subsets. A theory is weakly o-minimal if all its models are weakly o-minimal.
Note that, in contrast to o-minimality, it is possible for a theory to have models which are weakly o-minimal and to have other models which are not weakly o-minimal.
Difference from o-minimality.
In an o-minimal structure formula_1 the definable sets in formula_2 are finite unions of points and intervals, where "interval" stands for a sets of the form formula_3. For weakly o-minimal structures formula_1 this is relaxed so that the definable sets in "M" are finite unions of convex definable sets. A set formula_5 is convex if whenever "a" and "b" are in formula_5, "a" < "b" and "c" ∈  formula_2 satisfies that "a" < "c" < "b", then "c" is in "C". Points and intervals are of course convex sets, but there are convex sets which are not either points or intervals, as explained below.
If we have a weakly o-minimal structure expanding (R,<), the real ordered field, then the structure will be o-minimal. The two notions are different in other settings though. For example, let "R" be the ordered field of real algebraic numbers with the usual ordering < inherited from R. Take a transcendental number, say "π", and add a unary relation "S" to the structure given by the subset (−"π","π") ∩ "R". Now consider the subset "A" of "R" defined by the formula
Since we have a definable set that isn't a finite union of points and intervals, this structure is not o-minimal. However, it is known that the structure is weakly o-minimal, and in fact the theory of this structure is weakly o-minimal.