Weakly o-minimal types

We introduce and study weak o-minimality in the context of complete types in an arbitrary first-order theory. A type $p\in S\left(A\right)$ is weakly o-minimal if for some relatively A-definable linear order, <, on $p\left(C\right)$ every relatively $L_C$-definable subset of $p\left(C\right)$ has finitely many convex components in $\left(p\right(C),<)$. We establish many nice properties of weakly o-minimal types. For example, we prove that weakly o-minimal types are dp-minimal and share several properties of weight-one types in stable theories, and that a version of monotonicity theorem holds for relatively definable functions on the locus of a weakly o-minimal type.