Quasi-homeomorphisms, goldspectral spaces and jacspectral spaces

In this paper, we deal with the study of quasi-homeomorphisms, the Goldman prime spectrum and the Jacobson prime spectrum of a commutative ring. We prove that, if g:Y→X is a quasi-homeomorphism, Z a sober space and f:Y→Z a continuous map, then there exists a unique continuous map F:X→Z such that F o g = f. Let X be a T₀-space, q:X→^sX the injection of X onto its sobrification ^sX. It is shown, here, that q(Gold (X)) = Gold (^sX), where Gold (X) is the set of all locally closed points of X. Some applications are also indicated. The Jacobson prime spectrum of a commutative ring R is the set of all prime ideals of R which are intersections of some maximal ideals of R. One of our main results is a surprising answer to the problem of ordered disjoint union of jacspectral sets (ordered sets which are isomorphic to the Jacobson prime spectrum of some ring): Let {(X _λ, ≤_λ):λ ∈Λ} be a collection of ordered disjoint sets and X = U_λ∈Λ X_λ. Partially order X by declaring x ≤ y to mean that there exists λ ∈ Λ such that x, y ∈ X_λ and x ≤_λ y. Then the following statements are equivalent: (i) (X, ≤) is jacspectral. (ii) (X_λ, ≤_λ) is jacspectral, for each λ ∈ Λ.