Characterization of Scott–Isbell coincidence for coherent sources

Characterize the continuous dcpos L for which the Isbell and Scott topologies coincide on C(X,L) for every core-compact coherent topological space X.

Background

The paper establishes classifications of continuous dcpos according to Scott–Isbell coincidence for all core-compact sources, all compact core-compact sources, and all RW-spaces. It then considers the narrower class of core-compact coherent sources and defines SI(L) to mean coincidence for every source in that class.

The authors note that every continuous B-domain satisfies this coincidence property, but their obstruction method does not resolve the maximal target class. In particular, the Alexandrov detector used for the main separation argument is not coherent when its index set is infinite: two compact saturated subsets have an intersection covered by infinitely many open singletons with no finite subcover. Determining exactly which continuous dcpos satisfy coincidence for all core-compact coherent sources is therefore left unresolved.