Rocky Mountain Journal of Mathematics

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

Carmelo A. Finocchiaro, Marco Fontana, and Dario Spirito

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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 $D$.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 5 (2018), 1551-1583.

Dates
First available in Project Euclid: 19 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1539936036

Digital Object Identifier
doi:10.1216/RMJ-2018-48-5-1551

Mathematical Reviews number (MathSciNet)
MR3866559

Zentralblatt MATH identifier
06958792

Subjects
Primary: 13A10 13A15: Ideals; multiplicative ideal theory 13B10: Morphisms 13G05: Integral domains 14A05: Relevant commutative algebra [See also 13-XX] 54A10: Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) 54F65: Topological characterizations of particular spaces

Keywords
Spectral space spectral map Zariski topology constructible topology inverse topology hull-kernel topology stably compact space Smyth powerdomain co-compact topology de Groot duality upper Vietoris topology Scott topology closure operation semistar operation radical ideal ultrafilter topology

Citation

Finocchiaro, Carmelo A.; Fontana, Marco; Spirito, Dario. The upper Vietoris topology on the space of inverse-closed subsets of a spectral space and applications. Rocky Mountain J. Math. 48 (2018), no. 5, 1551--1583. doi:10.1216/RMJ-2018-48-5-1551. https://projecteuclid.org/euclid.rmjm/1539936036


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