{"title":"Hyperspaces of the double arrow","authors":"Sebastián Barría","doi":"10.1016/j.topol.2024.109125","DOIUrl":null,"url":null,"abstract":"<div><div>Let <span><math><mi>A</mi></math></span> and <span><math><mi>S</mi></math></span> denote the double arrow of Alexandroff and the Sorgenfrey line, respectively. We show that for any <span><math><mi>n</mi><mo>≥</mo><mn>1</mn></math></span>, the space of all unions of at most <em>n</em> closed intervals of <span><math><mi>A</mi></math></span> is not homogeneous. We also prove that the spaces of non-trivial convergent sequences of <span><math><mi>A</mi></math></span> and <span><math><mi>S</mi></math></span> are homogeneous. This partially solves an open question of A. Arhangel'skiǐ <span><span>[1]</span></span>. In contrast, we show that the space of closed intervals of <span><math><mi>S</mi></math></span> is homogeneous.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-10-29","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/S0166864124003109","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
Let and denote the double arrow of Alexandroff and the Sorgenfrey line, respectively. We show that for any , the space of all unions of at most n closed intervals of is not homogeneous. We also prove that the spaces of non-trivial convergent sequences of and are homogeneous. This partially solves an open question of A. Arhangel'skiǐ [1]. In contrast, we show that the space of closed intervals of is homogeneous.
让 A 和 S 分别表示亚历山大罗夫双箭线和索根弗雷线。我们证明，对于任意 n≥1，A 的最多 n 个封闭区间的所有联合的空间不是同质的。我们还证明了 A 和 S 的非琐收敛序列的空间是同质的。这部分解决了 A. Arhangel'skiǐ [1] 的一个未决问题。相反，我们证明了 S 的闭区间空间是同质的。
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.
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