Let $X$ be a topological space, and let $\operatorname{CL}(X)$ be its hyperspace. That is, $\operatorname{CL}(X)$ is the set of closed subsets of $X$, equipped with the minimal topology so that the canonical map $$x \mapsto \overline{\{x\}}$$is a homeomorphism onto its image.
Typically, the base space is assumed to be Hausdorff (or at least $T_1$), so that the closure of a singleton is the singleton itself. However, the definition of a hyperspace is perfectly suitable when the space is not Hausdorff, and surely this comes in handy sometimes.
What is a good example of a hyperspace $\operatorname{CL}(X)$ where the base space $X$ is non-Hausdorff?