When do the upper (a.k.a., lower Vietoris) and Scott topologies coincide on the Hoare hyperspace of a space?

I would like to talk about a nifty, recent result due to Yu Chen, Hui Kou, and Zhenchao Lyu. There are two natural topologies on the Hoare hyperspace of a space X, the Scott and the lower Vietoris topology, and one may wonder when they coincide. Outside of the realm of posets in their Scott topology, we will see that they rather rarely coincide. The result that is the core of this month’s post is that they do if X is a poset (with its Scott topology) satisfying what I will call the Chen-Kou-Lyu property; and that this property holds if the poset X is core-compact, or first-countable. Read the full post.