The complete lattice Lfan (part II)

Last time, we had started to study the complete lattice L_fan, namely just N × N, with equality as ordering on the first component and the usual ordering on the second component, plus an extra bottom element ⊥, plus an extra top element ⊤. This is a complete lattice with the Chen-Kou-Lyu property, and we have seen a few consequences of that. This time, we will see that L_fan is a maximal limit space (in particular, it is sober and weakly Hausdorff); we knew that it is not core-compact, but it is consonant; we knew that it is not first-countable, but its lattice of open sets is second-countable in its Scott topology; and the upper Vietoris and Scott topologies coincide on its Smyth hyperspace. Read the full post.