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