积分域上的马西亚斯拓扑学

IF 0.6 4区数学 Q3 MATHEMATICS Topology and its Applications Pub Date : 2024-09-10 DOI:10.1016/j.topol.2024.109070

Jhixon Macías

{"title":"积分域上的马西亚斯拓扑学","authors":"Jhixon Macías","doi":"10.1016/j.topol.2024.109070","DOIUrl":null,"url":null,"abstract":"<div><p>In this manuscript a recent topology on the positive integers generated by the collection of <span><math><mo>{</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>}</mo></math></span> where <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mo>=</mo><mo>{</mo><mi>m</mi><mo>:</mo><mi>gcd</mi><mo>⁡</mo><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>}</mo></math></span> is generalized over integral domains. Some of its topological properties are studied. Properties of this topology on infinite principal ideal domains that are not fields are also explored, and a new topological proof of the infinitude of prime elements is obtained (assuming the set of units is finite or not open), different from those presented in the style of H. Furstenberg. Finally, some problems are proposed.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"357 ","pages":"Article 109070"},"PeriodicalIF":0.6000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Macías topology on integral domains\",\"authors\":\"Jhixon Macías\",\"doi\":\"10.1016/j.topol.2024.109070\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this manuscript a recent topology on the positive integers generated by the collection of <span><math><mo>{</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>}</mo></math></span> where <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mo>=</mo><mo>{</mo><mi>m</mi><mo>:</mo><mi>gcd</mi><mo>⁡</mo><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>}</mo></math></span> is generalized over integral domains. Some of its topological properties are studied. Properties of this topology on infinite principal ideal domains that are not fields are also explored, and a new topological proof of the infinitude of prime elements is obtained (assuming the set of units is finite or not open), different from those presented in the style of H. Furstenberg. Finally, some problems are proposed.</p></div>\",\"PeriodicalId\":51201,\"journal\":{\"name\":\"Topology and its Applications\",\"volume\":\"357 \",\"pages\":\"Article 109070\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-09-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Topology and its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0166864124002554\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Topology and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166864124002554","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

在本手稿中，对由{σn:n∈N}集合（其中σn:={m:gcd(n,m)=1}）产生的正整数的最新拓扑学进行了积分域上的推广。研究了它的一些拓扑性质。此外，还探讨了这种拓扑在非域的无限主理想域上的性质，并得到了素元无穷大的新拓扑证明（假设单位集是有限的或不开放的），这与弗斯滕贝格（H. Furstenberg）风格的证明不同。最后，还提出了一些问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

The Macías topology on integral domains

In this manuscript a recent topology on the positive integers generated by the collection of ${σ_{n} : n \in N}$ where $σ_{n} : = {m : \gcd (n, m) = 1}$ is generalized over integral domains. Some of its topological properties are studied. Properties of this topology on infinite principal ideal domains that are not fields are also explored, and a new topological proof of the infinitude of prime elements is obtained (assuming the set of units is finite or not open), different from those presented in the style of H. Furstenberg. Finally, some problems are proposed.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.

期刊最新文献

Relatively functionally countable subsets of products Extendability to Marczewski-Burstin countably representable ideals MSNR spaces revisited On Ψω-factorizable groups On the functor of comonotonically maxitive functionals