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

TL;DR

This paper introduces a new topology on the set of inverse-closed subsets of a spectral space, proving it forms a spectral space and exploring its applications to algebraic structures like semistar operations.

Contribution

It defines a Zariski-like topology on inverse-closed subsets of spectral spaces, proves the space is spectral, and connects it to algebraic structures such as semistar operations.

Findings

The set of inverse-closed subsets forms a spectral space under the new topology.

The construction is functorial and extends the original spectral space.

Application to algebra shows homeomorphism with space of semistar operations.

Abstract

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 $D$.