When is ℒ(X) locally compact?

Let ℒ(X) be the dcpo of all lower semicontinuous maps from a space X to R₊ ∪ {∞}. We give it the Scott topology. Under which conditions on X is it locally compact? We will see that this happens exactly when X is core-compact, in two situations: when X is a poset with its Scott topology, and when X is an ⊙-consonant second countable space. I don’t know anything outside of these two cases. The case of posets has a remarkably short and clever proof, due to Xiaodong Jia. The case of ⊙-consonant second countable spaces has a much more tortuous proof, and relies on deep properties of hyperspaces discovered by Matthew de Brecht and Tatsuji Kawai. Read the full post.