The upper Vietoris topology on the space of inverse-closed subsets of a spectral space and applications

Given an arbitrary spectral space X, we consider the set X(X) of all nonempty subsets of X that are closed with respect to the inverse topology. We introduce a Zariski-like topology on X(X) and, after observing that it coincides the upper Vietoris topology, we prove that X(X) is itself a spectral space, that this construction is functorial, and that X(X) provides an extension of X in a more “complete” spectral space. Among the applications, we show that, starting from an integral domain D, X(Spec(D)) is homeomorphic to the (spectral) space of all the stable semistar operations of finite type on D.