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

Finocchiaro C. A.;
2018-01-01

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.
2018
Closure operation; Co-compact topology; Constructible topology; De groot duality; Hull-kernel topology; Inverse topology; Radical ideal; Scott topology; Semistar operation; Smyth powerdomain; Spectral map; Spectral space; Stably compact space; Ultrafilter topology; Upper Vietoris topology; Zariski topology
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11769/383269
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