{"title":"服部子空间","authors":"","doi":"10.1016/j.topol.2024.109077","DOIUrl":null,"url":null,"abstract":"<div><div>For a subset <em>A</em> of an almost topological group <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math></span>, the Hattori space <span><math><mi>H</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is a topological space whose underlying set is <em>G</em> and whose topology <span><math><mi>τ</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is defined as follows: if <span><math><mi>x</mi><mo>∈</mo><mi>A</mi></math></span> (respectively, <span><math><mi>x</mi><mo>∉</mo><mi>A</mi></math></span>), then the neighborhoods of <em>x</em> in <span><math><mi>H</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> are the same neighborhoods of <em>x</em> in the reflection group <span><math><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>,</mo><msub><mrow><mi>τ</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>)</mo></math></span> (respectively, <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math></span>). Given an infinite subset <em>X</em> of an almost topological group <em>G</em> and <span><math><mi>A</mi><mo>⊆</mo><mi>X</mi></math></span>, we denote by <span><math><mi>X</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>X</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span> and <em>X</em> to the spaces <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo>(</mo><mi>A</mi><mo>)</mo><msub><mrow><mo>|</mo></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo></math></span>, <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>τ</mi></mrow><mrow><mo>⁎</mo></mrow></msub><msub><mrow><mo>|</mo></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><msub><mrow><mo>|</mo></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo></math></span>, respectively. We say that <span><math><mi>X</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is the Hattori subspace associated to <em>A</em>. In this paper, we obtain information about Hattori subspaces. We show that some known topological spaces can be obtained as Hattori subspaces of some almost topological groups.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Hattori subspaces\",\"authors\":\"\",\"doi\":\"10.1016/j.topol.2024.109077\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>For a subset <em>A</em> of an almost topological group <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math></span>, the Hattori space <span><math><mi>H</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is a topological space whose underlying set is <em>G</em> and whose topology <span><math><mi>τ</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is defined as follows: if <span><math><mi>x</mi><mo>∈</mo><mi>A</mi></math></span> (respectively, <span><math><mi>x</mi><mo>∉</mo><mi>A</mi></math></span>), then the neighborhoods of <em>x</em> in <span><math><mi>H</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> are the same neighborhoods of <em>x</em> in the reflection group <span><math><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>,</mo><msub><mrow><mi>τ</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>)</mo></math></span> (respectively, <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>τ</mi><mo>)</mo></math></span>). Given an infinite subset <em>X</em> of an almost topological group <em>G</em> and <span><math><mi>A</mi><mo>⊆</mo><mi>X</mi></math></span>, we denote by <span><math><mi>X</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>X</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span> and <em>X</em> to the spaces <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo>(</mo><mi>A</mi><mo>)</mo><msub><mrow><mo>|</mo></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo></math></span>, <span><math><mo>(</mo><mi>X</mi><mo>,</mo><msub><mrow><mi>τ</mi></mrow><mrow><mo>⁎</mo></mrow></msub><msub><mrow><mo>|</mo></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><msub><mrow><mo>|</mo></mrow><mrow><mi>X</mi></mrow></msub><mo>)</mo></math></span>, respectively. We say that <span><math><mi>X</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is the Hattori subspace associated to <em>A</em>. In this paper, we obtain information about Hattori subspaces. We show that some known topological spaces can be obtained as Hattori subspaces of some almost topological groups.</div></div>\",\"PeriodicalId\":51201,\"journal\":{\"name\":\"Topology and its Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-10-15\",\"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/S0166864124002621\",\"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/S0166864124002621","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
对于一个几乎拓扑群(G,τ)的子集 A,服部空间 H(A) 是一个拓扑空间,其底层集是 G,其拓扑 τ(A) 的定义如下:如果 x∈A(分别为 x∉A),那么 x 在 H(A) 中的邻域就是 x 在反射群(G⁎,τ⁎)(分别为 (G,τ))中的邻域。给定几乎拓扑群 G 的无限子集 X 和 A⊆X,我们分别用 X(A)、X⁎ 和 X 表示空间 (X,τ(A)|X)、(X,τ⁎|X) 和 (X,τ|X)。我们说 X(A) 是与 A 相关联的服部子空间。我们证明,一些已知的拓扑空间可以作为一些近似拓扑群的服部子空间得到。
For a subset A of an almost topological group , the Hattori space is a topological space whose underlying set is G and whose topology is defined as follows: if (respectively, ), then the neighborhoods of x in are the same neighborhoods of x in the reflection group (respectively, ). Given an infinite subset X of an almost topological group G and , we denote by , and X to the spaces , and , respectively. We say that is the Hattori subspace associated to A. In this paper, we obtain information about Hattori subspaces. We show that some known topological spaces can be obtained as Hattori subspaces of some almost topological groups.
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.