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2018 The upper Vietoris topology on the space of inverse-closed subsets of a spectral space and applications
Carmelo A. Finocchiaro, Marco Fontana, Dario Spirito
Rocky Mountain J. Math. 48(5): 1551-1583 (2018). DOI: 10.1216/RMJ-2018-48-5-1551

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 $D$.

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Carmelo A. Finocchiaro. Marco Fontana. Dario Spirito. "The upper Vietoris topology on the space of inverse-closed subsets of a spectral space and applications." Rocky Mountain J. Math. 48 (5) 1551 - 1583, 2018. https://doi.org/10.1216/RMJ-2018-48-5-1551

Information

Published: 2018
First available in Project Euclid: 19 October 2018

zbMATH: 06958792
MathSciNet: MR3866559
Digital Object Identifier: 10.1216/RMJ-2018-48-5-1551

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Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.48 • No. 5 • 2018
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