Given an arbitrary spectral space X, we endow it with its specialization order ≤ and we study the interplay between suprema of subsets of (X,≤) and the constructible topology. More precisely, we examine when the supremum of a set Y ⊂ X exists and belongs to the constructible closure of Y. We apply such results to algebraic lattices of sets and to closure operations on them, proving density properties of some distinguished spaces of rings and ideals. Furthermore, we provide topological characterizations of some class of domains in terms of topological properties of their ideals.

The authors would like to express their sincere gratitude to the referee, whose remarkable suggestions helped to significantly improve the paper.