When is the frame of nuclei spatial: A new approach

Hofmann-Mislove through the Lenses of Priestley

“…6.4], if X is a spectral space, then the Hofmann-Mislove Theorem and Theorem 2.5 are simply reformulations of each other. Indeed, let D be a bounded distributive lattice and let L be the frame of ideals of D. Then L is a coherent frame [10, p. 64], where we recall that a frame L is coherent if the compact elements form a bounded sublattice of L that join-generates L. 1 In fact, sending D to L defines a covariant functor that establishes an equivalence between Dist and the category CohFrm of coherent frames and coherent morphisms (where a morphism is coherent if it is a frame homomorphism that sends compact elements to compact elements). Under this equivalence, the posets Filt(D) and OFilt(L) are isomorphic.…”

“…(2) Since Y is sober, Y is homeomorphic to the space of points of L (see, e.g., [12, p. 20]). Now apply (1).…”