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Involutions on the product of projective space and sphere 射影空间与球积的对合
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2025-06-19 DOI: 10.1016/j.topol.2025.109481
Dimpi , Hemant Kumar Singh
{"title":"Involutions on the product of projective space and sphere","authors":"Dimpi ,&nbsp;Hemant Kumar Singh","doi":"10.1016/j.topol.2025.109481","DOIUrl":"10.1016/j.topol.2025.109481","url":null,"abstract":"<div><div>This paper focuses on the actions of <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> on a finite CW-complex <em>X</em>, whose mod 2 cohomology is isomorphic to that of a product of projective space and sphere, <span><math><mi>F</mi><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>×</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span>, where <span><math><mi>F</mi></math></span> is either <span><math><mi>R</mi></math></span> or <span><math><mi>C</mi></math></span>. For the case when <em>X</em> is totally nonhomologous to zero in <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span>, we determine the possible connected fixed point sets and construct examples illustrating these possibilities. We also address the case when <em>X</em> is not totally nonhomologous to zero in <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span> under certain assumptions.</div><div>Furthermore, we compute the cohomology ring structure of the orbit spaces of free involutions on <em>X</em>. As an application, we derive the Borsuk-Ulam type results.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109481"},"PeriodicalIF":0.6,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144491641","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An extension of the class σE and complete Erdős space 扩展了类σE和完整的Erdős空间
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2025-06-19 DOI: 10.1016/j.topol.2025.109483
Alfredo Zaragoza
{"title":"An extension of the class σE and complete Erdős space","authors":"Alfredo Zaragoza","doi":"10.1016/j.topol.2025.109483","DOIUrl":"10.1016/j.topol.2025.109483","url":null,"abstract":"<div><div>In <span><span>[6]</span></span>, a topological characterization of the product of the rational numbers and complete Erdős space was found introducing the class <span><math><mi>σ</mi><mi>E</mi></math></span>. The objective of this work is to introduce the class of spaces <span><math><mi>σ</mi><msup><mrow><mi>E</mi></mrow><mrow><mo>−</mo></mrow></msup></math></span> and study some of its properties. We will also show that for certain subsets of complete Erdős space, the classes <span><math><mi>σ</mi><mi>E</mi></math></span> and <span><math><mi>σ</mi><msup><mrow><mi>E</mi></mrow><mrow><mo>−</mo></mrow></msup></math></span> coincide. This fact gives a partial answer to some questions of <span><span>[6]</span></span>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109483"},"PeriodicalIF":0.6,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144338271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Some variations of topological transitivity for CR-dynamical systems cr -动力系统拓扑传递性的一些变化
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2025-06-17 DOI: 10.1016/j.topol.2025.109482
Nayan Adhikary, Anima Nagar
{"title":"Some variations of topological transitivity for CR-dynamical systems","authors":"Nayan Adhikary,&nbsp;Anima Nagar","doi":"10.1016/j.topol.2025.109482","DOIUrl":"10.1016/j.topol.2025.109482","url":null,"abstract":"<div><div>We consider the topological dynamics of closed relations (CR) by studying one of the oldest dynamical property - ‘transitivity’. We investigate the two kinds of (closed relation) CR-dynamical systems - <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>G</mi><mo>)</mo></math></span> where the relation <span><math><mi>G</mi><mo>⊆</mo><mi>X</mi><mo>×</mo><mi>X</mi></math></span> is closed and <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>G</mi><mo>,</mo><mo>•</mo><mo>)</mo></math></span> giving the ‘suitable dynamics’ for a suitable closed relation <em>G</em>, where <em>X</em> is assumed to be a compact metric space without isolated points.</div><div><span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>G</mi><mo>)</mo></math></span> gives a general approach to study initial value problems for a set of initial conditions, whereas <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>G</mi><mo>,</mo><mo>•</mo><mo>)</mo></math></span> gives a general approach to study the dynamics of both continuous and quasi-continuous maps.</div><div>We observe that the dynamics of closed relations is richer than the dynamics of maps and find that we have much more versions of transitivity for these closed relations than what is known for maps.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109482"},"PeriodicalIF":0.6,"publicationDate":"2025-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144314018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Groups from walks on countable ordinals 在可数序数上的群
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2025-06-13 DOI: 10.1016/j.topol.2025.109480
Stepan Milošević , Stevo Todorčević
{"title":"Groups from walks on countable ordinals","authors":"Stepan Milošević ,&nbsp;Stevo Todorčević","doi":"10.1016/j.topol.2025.109480","DOIUrl":"10.1016/j.topol.2025.109480","url":null,"abstract":"<div><div>This paper deals with topological groups constructed from characteristics of walks on countable ordinals. We introduce two groups, constructed from <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, and show they are both Fréchet, non-metrizable and that the topological group constructed from <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> has the maximal Tukey type among topological groups of character <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>. We also consider the group associated with <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> introduced in [12] and show that under some set-theoretic assumptions the group does not have the maximal Tukey type.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109480"},"PeriodicalIF":0.6,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Definable quotients in locally o-minimal structures 局部0极小结构中的可定义商
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2025-06-11 DOI: 10.1016/j.topol.2025.109479
Masato Fujita , Tomohiro Kawakami
{"title":"Definable quotients in locally o-minimal structures","authors":"Masato Fujita ,&nbsp;Tomohiro Kawakami","doi":"10.1016/j.topol.2025.109479","DOIUrl":"10.1016/j.topol.2025.109479","url":null,"abstract":"<div><div>Let <span><math><mi>F</mi><mo>=</mo><mo>(</mo><mi>F</mi><mo>,</mo><mo>+</mo><mo>.</mo><mo>⋅</mo><mo>,</mo><mo>&lt;</mo><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>)</mo></math></span> be a definably complete locally o-minimal expansion of an ordered field. We demonstrate the existence of definable quotients of definable sets by definable equivalence relations when several technical conditions are satisfied. These conditions are satisfied when <em>X</em> is a locally closed definable subset of <span><math><msup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> and there is a definable proper action of a definable topological group <em>G</em> on <em>X</em>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109479"},"PeriodicalIF":0.6,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144279525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Products of ordinal-like spaces 类序数空间的积
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2025-06-11 DOI: 10.1016/j.topol.2025.109478
Jocelyn R. Bell
{"title":"Products of ordinal-like spaces","authors":"Jocelyn R. Bell","doi":"10.1016/j.topol.2025.109478","DOIUrl":"10.1016/j.topol.2025.109478","url":null,"abstract":"<div><div>We consider spaces that resemble ordinal spaces and products of these spaces, in particular Σ-products and uniform products: the topology generated by the uniformity of uniform convergence on the set of all functions from <span><math><mi>N</mi></math></span> to the space. We play a weak variation of the proximal infinite game on these spaces to conclude that they are countably proximal and so both pseudonormal and <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> (in the sense of Arhangel'skii). In particular, it follows from our results that a Σ-product of ordinal spaces is pseudonormal.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109478"},"PeriodicalIF":0.6,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144291087","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Cauchy sequences in b-metric spaces b-度量空间中的柯西序列
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2025-06-09 DOI: 10.1016/j.topol.2025.109477
Lech Pasicki
{"title":"Cauchy sequences in b-metric spaces","authors":"Lech Pasicki","doi":"10.1016/j.topol.2025.109477","DOIUrl":"10.1016/j.topol.2025.109477","url":null,"abstract":"<div><div>In the present paper a criterion for Cauchy sequences in b-metric spaces is given. To show how it works, an analogon of the Meir-Keeler fixed point theorem is proved. Consequently, an extension of the well-known theorems of Czerwik and of Matkowski is presented.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109477"},"PeriodicalIF":0.6,"publicationDate":"2025-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144242484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Asymptotic Betti numbers of random subcomplexes 随机子复的渐近Betti数
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2025-06-06 DOI: 10.1016/j.topol.2025.109476
Nermin Salepci, Jean-Yves Welschinger
{"title":"Asymptotic Betti numbers of random subcomplexes","authors":"Nermin Salepci,&nbsp;Jean-Yves Welschinger","doi":"10.1016/j.topol.2025.109476","DOIUrl":"10.1016/j.topol.2025.109476","url":null,"abstract":"<div><div>We prove that the normalized expected Betti numbers of a random subcomplex in the <em>d</em>-th barycentric subdivision of a finite simplicial complex converge to universal limits as <em>d</em> grows to +∞.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109476"},"PeriodicalIF":0.6,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144255193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Weight, net weight, and elementary submodels 权重、净重和基本子模型
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2025-06-02 DOI: 10.1016/j.topol.2025.109469
Alan Dow , István Juhász
{"title":"Weight, net weight, and elementary submodels","authors":"Alan Dow ,&nbsp;István Juhász","doi":"10.1016/j.topol.2025.109469","DOIUrl":"10.1016/j.topol.2025.109469","url":null,"abstract":"<div><div>In this note we prove several theorems that are related to some results and problems from <span><span>[6]</span></span>.</div><div>We answer two of the main questions that were raised in <span><span>[6]</span></span>. First we give a ZFC example of a <em>Hausdorff</em> space in <span><math><mi>C</mi><mo>(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> that has uncountable net weight. Then we prove that after adding any number of Cohen reals to a model of CH, in the extension every <em>regular</em> space in <span><math><mi>C</mi><mo>(</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span> has countable net weight.</div><div>In the last section we prove in ZFC the following two statements:</div><div>(i) If <span><math><mi>S</mi><mo>⊂</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is stationary then for any <em>regular</em> topology on <em>S</em> of uncountable weight  <em>S</em> has a non-stationary subset that has uncountable weight as well.</div><div>(ii) For any topology on <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, if all final segments of <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> have uncountable weight then <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> has a non-stationary subset of uncountable weight.</div><div>In contrast to this, it was shown in <span><span>[6]</span></span> that the analogous statements for net weight are not provable in ZFC.</div><div>It is remarkable that all our proofs of the above results make essential use of elementary submodels.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109469"},"PeriodicalIF":0.6,"publicationDate":"2025-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144204225","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A generating set of Reidemeister moves of oriented virtual knots 有向虚节的瑞德迈斯特移动生成集
IF 0.6 4区 数学
Topology and its Applications Pub Date : 2025-05-30 DOI: 10.1016/j.topol.2025.109468
Danish Ali
{"title":"A generating set of Reidemeister moves of oriented virtual knots","authors":"Danish Ali","doi":"10.1016/j.topol.2025.109468","DOIUrl":"10.1016/j.topol.2025.109468","url":null,"abstract":"<div><div>In oriented knot theory, verifying a quantity is an invariant involves checking its invariance under all oriented Reidemeister moves, a process that can be intricate and time-consuming. A generating set of oriented moves simplifies this by requiring verification for only a minimal subset from which all other moves can be derived. While generating sets for classical oriented Reidemeister moves are well-established, their virtual counterparts are less explored. In this study, we enumerate the oriented virtual Reidemeister moves, identifying seventeen distinct moves after accounting for redundancies due to rotational and combinatorial symmetries. We prove that a four-element subset serves as a generating set for these moves. This result offers a streamlined approach to verifying invariants of oriented virtual knots and lays the groundwork for future advancements in virtual knot theory, particularly in the study of invariants and their computational properties.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"373 ","pages":"Article 109468"},"PeriodicalIF":0.6,"publicationDate":"2025-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144196488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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