关于对角线度和星形网络

IF 0.6 4区 数学 Q3 MATHEMATICS Topology and its Applications Pub Date : 2024-10-01 DOI:10.1016/j.topol.2024.109074
Nathan Carlson
{"title":"关于对角线度和星形网络","authors":"Nathan Carlson","doi":"10.1016/j.topol.2024.109074","DOIUrl":null,"url":null,"abstract":"<div><div>Given an open cover <span><math><mi>U</mi></math></span> of a topological space <em>X</em>, we introduce the notion of a star network for <span><math><mi>U</mi></math></span>. The associated cardinal function <span><math><mi>s</mi><mi>n</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>, where <span><math><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mi>s</mi><mi>n</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>, is used to establish new cardinal inequalities involving diagonal degrees. We show <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><mi>s</mi><mi>n</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>Δ</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> for a <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> space <em>X</em>, giving a partial answer to a long-standing question of Angelo Bella. Many further results are given using variations of <span><math><mi>s</mi><mi>n</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. One result has as corollaries Buzyakova's theorem that a ccc space with a regular <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>δ</mi></mrow></msub></math></span>-diagonal has cardinality at most <span><math><mi>c</mi></math></span>, as well as three results of Gotchev. Further results lead to logical improvements of theorems of Basile, Bella, and Ridderbos, a partial solution to a question of the same authors, and a theorem of Gotchev, Tkachenko, and Tkachuk. Finally, we define the Urysohn extent <span><math><mi>U</mi><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> with the property <span><math><mi>U</mi><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mi>min</mi><mo>⁡</mo><mo>{</mo><mi>a</mi><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>}</mo></math></span> and use the Erdős-Rado theorem to show that <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>U</mi><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo><mover><mrow><mi>Δ</mi></mrow><mo>‾</mo></mover><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> for any Urysohn space <em>X</em>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"357 ","pages":"Article 109074"},"PeriodicalIF":0.6000,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On diagonal degrees and star networks\",\"authors\":\"Nathan Carlson\",\"doi\":\"10.1016/j.topol.2024.109074\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>Given an open cover <span><math><mi>U</mi></math></span> of a topological space <em>X</em>, we introduce the notion of a star network for <span><math><mi>U</mi></math></span>. The associated cardinal function <span><math><mi>s</mi><mi>n</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>, where <span><math><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mi>s</mi><mi>n</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>, is used to establish new cardinal inequalities involving diagonal degrees. We show <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><mi>s</mi><mi>n</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>Δ</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> for a <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> space <em>X</em>, giving a partial answer to a long-standing question of Angelo Bella. Many further results are given using variations of <span><math><mi>s</mi><mi>n</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. One result has as corollaries Buzyakova's theorem that a ccc space with a regular <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>δ</mi></mrow></msub></math></span>-diagonal has cardinality at most <span><math><mi>c</mi></math></span>, as well as three results of Gotchev. Further results lead to logical improvements of theorems of Basile, Bella, and Ridderbos, a partial solution to a question of the same authors, and a theorem of Gotchev, Tkachenko, and Tkachuk. Finally, we define the Urysohn extent <span><math><mi>U</mi><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> with the property <span><math><mi>U</mi><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mi>min</mi><mo>⁡</mo><mo>{</mo><mi>a</mi><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>}</mo></math></span> and use the Erdős-Rado theorem to show that <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>U</mi><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo><mover><mrow><mi>Δ</mi></mrow><mo>‾</mo></mover><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> for any Urysohn space <em>X</em>.</div></div>\",\"PeriodicalId\":51201,\"journal\":{\"name\":\"Topology and its Applications\",\"volume\":\"357 \",\"pages\":\"Article 109074\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-10-01\",\"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/S0166864124002591\",\"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/S0166864124002591","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

给定拓扑空间 X 的开盖 U,我们引入 U 的星形网络概念。相关的心形函数 sn(X)(其中 e(X)≤sn(X)≤L(X) )用于建立涉及对角度的新心形不等式。我们证明了 T1 空间 X 的 |X|≤sn(X)Δ(X),部分回答了安杰洛-贝拉(Angelo Bella)的一个长期问题。利用 sn(X) 的变化给出了许多进一步的结果。其中一个结果的推论是布扎科娃(Buzyakova)的定理,即具有规则 Gδ 对角线的ccc 空间的心性至多为 c,以及哥切夫(Gotchev)的三个结果。进一步的结果导致了巴西尔、贝拉和里德博斯定理的逻辑改进,同一作者的一个问题的部分解答,以及哥切夫、特卡琴科和特卡丘克的一个定理。最后,我们定义了具有 Ue(X)≤min{aL(X),e(X)} 特性的乌里索恩程度 Ue(X),并使用厄尔多斯-拉多定理证明了对于任何乌里索恩空间 X,|X|≤2Ue(X)Δ‾(X)。
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
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
On diagonal degrees and star networks
Given an open cover U of a topological space X, we introduce the notion of a star network for U. The associated cardinal function sn(X), where e(X)sn(X)L(X), is used to establish new cardinal inequalities involving diagonal degrees. We show |X|sn(X)Δ(X) for a T1 space X, giving a partial answer to a long-standing question of Angelo Bella. Many further results are given using variations of sn(X). One result has as corollaries Buzyakova's theorem that a ccc space with a regular Gδ-diagonal has cardinality at most c, as well as three results of Gotchev. Further results lead to logical improvements of theorems of Basile, Bella, and Ridderbos, a partial solution to a question of the same authors, and a theorem of Gotchev, Tkachenko, and Tkachuk. Finally, we define the Urysohn extent Ue(X) with the property Ue(X)min{aL(X),e(X)} and use the Erdős-Rado theorem to show that |X|2Ue(X)Δ(X) for any Urysohn space X.
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
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
Editorial Board The Rudin-Kiesler pre-order and the Pixley-Roy spaces over ultrafilters On Dold-Whitney's parallelizability of 4-manifolds The uniform convergence topology on separable subsets Relatively functionally countable subsets of products
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1