{"title":"On diagonal degrees and star networks","authors":"","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":null,"pages":null},"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}
引用次数: 0
Abstract
Given an open cover of a topological space X, we introduce the notion of a star network for . The associated cardinal function , where , is used to establish new cardinal inequalities involving diagonal degrees. We show for a space X, giving a partial answer to a long-standing question of Angelo Bella. Many further results are given using variations of . One result has as corollaries Buzyakova's theorem that a ccc space with a regular -diagonal has cardinality at most , 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 with the property and use the Erdős-Rado theorem to show that for any Urysohn space X.
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.