When is ℒ(X) locally compact? II. The non-consonant case and the non-second-countable case

Last month, we have seen that ℒ(X) is locally compact in its Scott topology (equivalently, core-compact, equivalently, stably compact) if and only if OX is locally compact in its Scott topology (equivalently, core-compact, equivalently, stably compact), and that those properties are equivalent to X being core-compact, provided that X is a poset with its Scott topology or a ⊙-consonant second-countable space. Matthew de Brecht observed that the latter condition can be relaxed. We will see that we only need X to be second-countable or consonant for the equivalence to hold. Read the full post.