  We introduce and study weak o-minimality in the context of complete types in
an arbitrary first-order theory. A type $p\in S(A)$ is weakly o-minimal if for
some relatively $A$-definable linear order, $<$, on $p(\mathfrak{C})$ every
relatively $L_{\mathfrak{C}}$-definable subset of $p(\mathfrak{C})$ has
finitely many convex components in $(p(\mathfrak{C}),<)$. 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.
