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 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.

The Zariski topology on sets of semistar operations without finite-type assumptions

“…as Ω ranges among the compact open subsets of X. This topology coincides with the upper Vietoris topology [11,Proposition 3.1], and in particular it makes X (X) a spectral space [11,Theorem 3.4 Under this topology, both Over(D) [8,Proposition 3.5] and Zar(D) [6,7] are spectral spaces.…”

“…Let X (X) be the set of the nonempty subsets of X that are closed in the inverse topology. In [11] and [9, Section 4], X (X) was endowed with a natural topology, defined by taking, as a subbasis of open sets, the sets of the form…”

Topological properties of localizations, flat overrings and sublocalizations

“…The space X inv is again a spectral space. Following [15], we denote by X (X) the space of nonempty subsets of X that are closed in the inverse topology; this space can be endowed with a topology having, as a basis of open sets, the sets of the form…”

“…as Ω ranges among the open and compact subspaces of X. Under this topology, X (X) is again a spectral space [15,Theorem 3.2(1)].…”

“…In the induced topology, both the space SStar f (D) of finite-type operations and the space SStar f,sp (D) of finite-type spectral operations are spectral (see [16,Theorem 2.13] for the former and [13,Theorem 4.6] for the latter). Moreover, SStar f,sp (D) is homeomorphic to X (D) [15,Proposition 5.2]. The t-operation is the finite-type operation associated to the voperation; that is, t := v f .…”