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

Abstract

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