Given a theory $T$, say that a formula $\varphi(x,y)$ is downward directed if, $T \vdash \forall y_1 \forall y_2 \exists y_3 \forall x (\varphi(x,y_3)\rightarrow (\varphi(x,y_1)\wedge \varphi(x,y_2))$. We present a theorem stating vaguely that, in an o-minimal theory, any parametrically definable set of formulas that extends to a (complete global) definable type admits a refinement to a downward directed formula. This follows from o-minimal cell decomposition. We explain how this result can be used to prove the equivalence between two known notions of topological compactness for definable topologies in o-minimal structures. We then move on to discuss this theorem in the context of weakly o-minimal and other distal and tame topological settings. In particular, we discuss distal cell decomposition in connection with the theorem.