The Basic Zariski Topology

Q4 Mathematics

Confluentes Mathematici Pub Date : 2016-02-03 DOI:10.5802/CML.18

Davide Rinaldi, G. Sambin, P. Schuster

引用次数: 7

Abstract

We present the Zariski spectrum as an inductively generated basic topology à la Martin-Löf and Sambin. Since we can thus get by without considering powers and radicals, this simplifies the presentation as a formal topology initiated by Sigstam. Our treatment includes closed subspaces and basic opens: that is, arbitrary quotients and singleton localisations. All the effective objects under consideration are introduced by means of inductive definitions. The notions of spatiality and reducibility are characterized for the class of Zariski formal topologies, and their nonconstructive content is pointed out: while spatiality implies classical logic, reducibility corresponds to a fragment of the Axiom of Choice in the form of Russell’s Multiplicative Axiom.

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基本的Zariski拓扑

我们将Zariski谱表示为归纳生成的基本拓扑(如Martin-Löf和Sambin)。由于我们可以不考虑幂和根，这简化了作为Sigstam发起的形式化拓扑的表示。我们的处理包括闭子空间和基本开:即任意商和单态定位。所考虑的所有有效对象都是通过归纳定义引入的。对Zariski形式拓扑类的空间性和可约性概念进行了表征，并指出了它们的非建构性内容:空间性暗示经典逻辑，而可约性对应于罗素乘法公理形式的选择公理片段。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Confluentes Mathematici Mathematics-Mathematics (miscellaneous)

CiteScore

0.60

自引率

0.00%

发文量

期刊介绍： Confluentes Mathematici is a mathematical research journal. Since its creation in 2009 by the Institut Camille Jordan UMR 5208 and the Unité de Mathématiques Pures et Appliquées UMR 5669 of the Université de Lyon, it reflects the wish of the mathematical community of Lyon—Saint-Étienne to participate in the new forms of scientific edittion. The journal is electronic only, fully open acces and without author charges. The journal aims to publish high quality mathematical research articles in English, French or German. All domains of Mathematics (pure and applied) and Mathematical Physics will be considered, as well as the History of Mathematics. Confluentes Mathematici also publishes survey articles. Authors are asked to pay particular attention to the expository style of their article, in order to be understood by all the communities concerned.