Pub Date : 2024-08-26DOI: 10.1016/j.topol.2024.109050
Hualin Miao , Longchun Wang , Qingguo Li
In this paper, we study the D-completion, well-filterification and sobrification of a space. First, we present an example of a tapered closed set which is neither the closure of a directed set nor a closed KF-set. In 2020, Xu et al. asked whether closed RD-sets are exactly closed WD-sets for every space. This example also gives a negative answer to the above problem, since each tapered closed set is a closed WD-set. Second, we provide a direct characterization for the D-completion of a poset by using the notion of pre-C-compact elements. Finally, for a given space, we give some sufficient conditions which guarantee that each pair of its standard D-completion, standard well-filterification and standard sobrification agrees.
{"title":"D-completion, well-filterification and sobrification","authors":"Hualin Miao , Longchun Wang , Qingguo Li","doi":"10.1016/j.topol.2024.109050","DOIUrl":"10.1016/j.topol.2024.109050","url":null,"abstract":"<div><p>In this paper, we study the <em>D</em>-completion, well-filterification and sobrification of a <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> space. First, we present an example of a tapered closed set which is neither the closure of a directed set nor a closed <em>KF</em>-set. In 2020, Xu et al. asked whether closed <em>RD</em>-sets are exactly closed <em>WD</em>-sets for every <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> space. This example also gives a negative answer to the above problem, since each tapered closed set is a closed <em>WD</em>-set. Second, we provide a direct characterization for the <em>D</em>-completion of a poset by using the notion of pre-<em>C</em>-compact elements. Finally, for a given <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> space, we give some sufficient conditions which guarantee that each pair of its standard <em>D</em>-completion, standard well-filterification and standard sobrification agrees.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"356 ","pages":"Article 109050"},"PeriodicalIF":0.6,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142130076","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}
Pub Date : 2024-08-22DOI: 10.1016/j.topol.2024.109048
Franco Barragán , Anahí Rojas , Jesús F. Tenorio
<div><p>Let <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> be a dynamical system, where <em>X</em> is a nondegenerate continuum and <em>f</em> is a map. For any positive integer <em>n</em>, we consider the hyperspace <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> with the Vietoris topology. For <span><math><mi>n</mi><mo>></mo><mn>1</mn></math></span> and <span><math><mi>K</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> the subset <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>,</mo><mi>X</mi><mo>)</mo></math></span> of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is defined as the collection of elements of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> containing <em>K</em>. We consider the quotient hyperspace <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mi>⧸</mi><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>,</mo><mi>X</mi><mo>)</mo></math></span>, which is obtained from <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> by shrinking <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>,</mo><mi>X</mi><mo>)</mo></math></span> to one point set. Furthermore, we consider the induced maps <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo><mo>:</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>→</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>f</mi><mo>)</mo><mo>:</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo><mo>→</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. In this paper, we introduce the dynamical system <span><math><mo>(</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>f</mi><mo>)</mo><mo>)</mo></math></span> and we study relationships between the conditions <span><math><mi>f</mi><mo>∈</mo><mi>M</mi></math></span>, <span><math><msub><mrow><mi>F</mi></mr
{"title":"Dynamic properties of the dynamical system (FnK(X),FnK(f))","authors":"Franco Barragán , Anahí Rojas , Jesús F. Tenorio","doi":"10.1016/j.topol.2024.109048","DOIUrl":"10.1016/j.topol.2024.109048","url":null,"abstract":"<div><p>Let <span><math><mo>(</mo><mi>X</mi><mo>,</mo><mi>f</mi><mo>)</mo></math></span> be a dynamical system, where <em>X</em> is a nondegenerate continuum and <em>f</em> is a map. For any positive integer <em>n</em>, we consider the hyperspace <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> with the Vietoris topology. For <span><math><mi>n</mi><mo>></mo><mn>1</mn></math></span> and <span><math><mi>K</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> the subset <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>,</mo><mi>X</mi><mo>)</mo></math></span> of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is defined as the collection of elements of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> containing <em>K</em>. We consider the quotient hyperspace <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mi>⧸</mi><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>,</mo><mi>X</mi><mo>)</mo></math></span>, which is obtained from <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> by shrinking <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>,</mo><mi>X</mi><mo>)</mo></math></span> to one point set. Furthermore, we consider the induced maps <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo><mo>:</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>→</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>f</mi><mo>)</mo><mo>:</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo><mo>→</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. In this paper, we introduce the dynamical system <span><math><mo>(</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>K</mi></mrow></msubsup><mo>(</mo><mi>f</mi><mo>)</mo><mo>)</mo></math></span> and we study relationships between the conditions <span><math><mi>f</mi><mo>∈</mo><mi>M</mi></math></span>, <span><math><msub><mrow><mi>F</mi></mr","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"356 ","pages":"Article 109048"},"PeriodicalIF":0.6,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142098390","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}
Pub Date : 2024-08-22DOI: 10.1016/j.topol.2024.109047
Oleksandr Maslyuchenko , Vadym Myronyk , Roman Ivasiuk
We consider two natural topologies on the space of all separately continuous functions defined on the product of two topological spaces X and Y and ranged into a topological or metric space Z. These topologies are the cross-open topology and the cross-uniform topology. We show that these topologies coincides if X and Y are pseudocompacts and Z is a metric space. We prove that a compact space K embeds into for infinite compacts X, Y and a metrizable space if and only if the weight of K is less than the sharp cellularity of both spaces X and Y.
我们考虑了空间 S（X×Y,Z）上的两个自然拓扑，S（X×Y,Z）是定义在两个拓扑空间 X 和 Y 的乘积上的所有独立连续函数，并被置换到一个拓扑或度量空间 Z 中。我们证明，如果 X 和 Y 是伪紧凑且 Z 是度量空间，这些拓扑就会重合。我们证明，对于无限紧凑的 X、Y 和可元空间 Z⊇R，当且仅当 K 的权重小于 X 和 Y 两个空间的锐胞度时，紧凑空间 K 嵌入到 S(X×Y,Z)中。
{"title":"Compact subspaces of the space of separately continuous functions with the cross-uniform topology","authors":"Oleksandr Maslyuchenko , Vadym Myronyk , Roman Ivasiuk","doi":"10.1016/j.topol.2024.109047","DOIUrl":"10.1016/j.topol.2024.109047","url":null,"abstract":"<div><p>We consider two natural topologies on the space <span><math><mi>S</mi><mo>(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo>)</mo></math></span> of all separately continuous functions defined on the product of two topological spaces <em>X</em> and <em>Y</em> and ranged into a topological or metric space <em>Z</em>. These topologies are the cross-open topology and the cross-uniform topology. We show that these topologies coincides if <em>X</em> and <em>Y</em> are pseudocompacts and <em>Z</em> is a metric space. We prove that a compact space <em>K</em> embeds into <span><math><mi>S</mi><mo>(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo>)</mo></math></span> for infinite compacts <em>X</em>, <em>Y</em> and a metrizable space <span><math><mi>Z</mi><mo>⊇</mo><mi>R</mi></math></span> if and only if the weight of <em>K</em> is less than the sharp cellularity of both spaces <em>X</em> and <em>Y</em>.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"356 ","pages":"Article 109047"},"PeriodicalIF":0.6,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0166864124002323/pdfft?md5=5bdb3be46d45742e443d43a7c083ed49&pid=1-s2.0-S0166864124002323-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142098389","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-22DOI: 10.1016/j.topol.2024.109046
Meng Bao, Xiaolan Liu
A topological gyrogroup is a gyrogroup endowed with a topology such that the binary operation is jointly continuous and the inverse mapping is also continuous. It is shown that a strongly topological gyrogroup is a q-space if and only if it is an M-space. Then a characterization about weakly feathered strongly topological gyrogroups is given, that is, a strongly topological gyrogroup G is weakly feathered if and only if it contains a compact strong subgyrogroup H such that the quotient space is submetrizable.
拓扑陀螺群是具有拓扑结构的陀螺群，其二元操作是连续的，反映射也是连续的。研究表明，强拓扑陀螺群是一个 q 空间，当且仅当它是一个 M 空间。然后给出了一个关于弱羽化强拓扑陀螺群的特征，即强拓扑陀螺群 G 是弱羽化的，当且仅当它包含一个紧凑的强子陀螺群 H，使得商空间 G/H 是可亚对称的。
{"title":"The generalized metric property in strongly topological gyrogroups","authors":"Meng Bao, Xiaolan Liu","doi":"10.1016/j.topol.2024.109046","DOIUrl":"10.1016/j.topol.2024.109046","url":null,"abstract":"<div><p>A topological gyrogroup is a gyrogroup endowed with a topology such that the binary operation is jointly continuous and the inverse mapping is also continuous. It is shown that a strongly topological gyrogroup is a <em>q</em>-space if and only if it is an <em>M</em>-space. Then a characterization about weakly feathered strongly topological gyrogroups is given, that is, a strongly topological gyrogroup <em>G</em> is weakly feathered if and only if it contains a compact strong subgyrogroup <em>H</em> such that the quotient space <span><math><mi>G</mi><mo>/</mo><mi>H</mi></math></span> is submetrizable.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"356 ","pages":"Article 109046"},"PeriodicalIF":0.6,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142084147","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}
Pub Date : 2024-08-20DOI: 10.1016/j.topol.2024.109044
Pierre Matet
We revisit the application of Shelah's Revised GCH Theorem [19] to diamond. We also formulate a generalization of the theorem and prove a small fragment of it. Finally we consider another application of the theorem, to covering numbers of the form .
{"title":"Meeting, covering and Shelah's Revised GCH","authors":"Pierre Matet","doi":"10.1016/j.topol.2024.109044","DOIUrl":"10.1016/j.topol.2024.109044","url":null,"abstract":"<div><p>We revisit the application of Shelah's Revised GCH Theorem <span><span>[19]</span></span> to diamond. We also formulate a generalization of the theorem and prove a small fragment of it. Finally we consider another application of the theorem, to covering numbers of the form <span><math><mrow><mi>cov</mi></mrow><mo>(</mo><mo>−</mo><mo>,</mo><mo>−</mo><mo>,</mo><mo>−</mo><mo>,</mo><mi>ω</mi><mo>)</mo></math></span>.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"356 ","pages":"Article 109044"},"PeriodicalIF":0.6,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142087191","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}
Pub Date : 2024-08-20DOI: 10.1016/j.topol.2024.109042
Erica Flapan , Hugh Howards
Menasco proved that if G is a reduced, alternating, connected diagram of a link L and G is prime then L is prime. This surprising and important result has been generalized to other classes of links, as well as to tangles and spatial graphs. After exploring some issues with previous results, we obtain new splitting results for tangles and spatial graphs.
梅纳斯科证明，如果 G 是链接 L 的还原、交替、连通图，并且 G 是素数，那么 L 就是素数。这一惊人而重要的结果已被推广到其他类别的链接以及缠结和空间图。在探讨了之前结果的一些问题之后，我们得到了缠结图和空间图表的新分裂结果。
{"title":"Splittings of tangles and spatial graphs","authors":"Erica Flapan , Hugh Howards","doi":"10.1016/j.topol.2024.109042","DOIUrl":"10.1016/j.topol.2024.109042","url":null,"abstract":"<div><p>Menasco proved that if <em>G</em> is a reduced, alternating, connected diagram of a link <em>L</em> and <em>G</em> is prime then <em>L</em> is prime. This surprising and important result has been generalized to other classes of links, as well as to tangles and spatial graphs. After exploring some issues with previous results, we obtain new splitting results for tangles and spatial graphs.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"356 ","pages":"Article 109042"},"PeriodicalIF":0.6,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S016686412400227X/pdfft?md5=f1c541fd08540275d44340e861b55e7f&pid=1-s2.0-S016686412400227X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142095139","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-16DOI: 10.1016/j.topol.2024.109045
Nursultan Kuanyshov
We define and develop a homotopy invariant notion for the sequential topological complexity of a map , denoted , that interacts with and in the same way Jamie Scott's topological complexity map interacts with and . Furthermore, we apply to studying group homomorphisms .
In addition, we give the characterization of cohomological dimension of group homomorphisms.
{"title":"On the sequential topological complexity of group homomorphisms","authors":"Nursultan Kuanyshov","doi":"10.1016/j.topol.2024.109045","DOIUrl":"10.1016/j.topol.2024.109045","url":null,"abstract":"<div><p>We define and develop a homotopy invariant notion for the sequential topological complexity of a map <span><math><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></math></span>, denoted <span><math><mi>T</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span>, that interacts with <span><math><mi>T</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and <span><math><mi>T</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>Y</mi><mo>)</mo></math></span> in the same way Jamie Scott's topological complexity map <span><math><mi>T</mi><mi>C</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></span> interacts with <span><math><mi>T</mi><mi>C</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> and <span><math><mi>T</mi><mi>C</mi><mo>(</mo><mi>Y</mi><mo>)</mo></math></span>. Furthermore, we apply <span><math><mi>T</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span> to studying group homomorphisms <span><math><mi>ϕ</mi><mo>:</mo><mi>Γ</mi><mo>→</mo><mi>Λ</mi></math></span>.</p><p>In addition, we give the characterization of cohomological dimension of group homomorphisms.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"356 ","pages":"Article 109045"},"PeriodicalIF":0.6,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142098388","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}
Pub Date : 2024-08-14DOI: 10.1016/j.topol.2024.109043
Zhongjian Zhu, Tian Jin
In this paper, we develop the new method to compute the homotopy groups of the mapping cone beyond the metastable range by analysing the homotopy of the n-th filtration of the relative James construction for CW-pair , defined by B. Gray, which is homotopy equivalent to the homotopy fiber of the pinch map . As an application, we compute the 5 and 6-dim unstable homotopy groups of 3-dimensional mod Moore spaces for all positive integers r.
在本文中，我们通过分析 B. 格雷定义的 CW 对 AiX 的相对詹姆斯构造 J(X,A) 的 n-th 滤波的同调，发展了计算映射锥 Cf=Y∪fCX 超越可变范围的同调群的新方法，该方法等同于捏合映射 X∪iCA→ΣA 的同调纤维。作为应用，我们计算了所有正整数 r 的 3 维 mod 2r 摩尔空间的 5 维和 6 维不稳定同调群。
{"title":"The relative James construction and its application to homotopy groups","authors":"Zhongjian Zhu, Tian Jin","doi":"10.1016/j.topol.2024.109043","DOIUrl":"10.1016/j.topol.2024.109043","url":null,"abstract":"<div><p>In this paper, we develop the new method to compute the homotopy groups of the mapping cone <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mi>Y</mi><msub><mrow><mo>∪</mo></mrow><mrow><mi>f</mi></mrow></msub><mi>C</mi><mi>X</mi></math></span> beyond the metastable range by analysing the homotopy of the <em>n</em>-th filtration of the relative James construction <span><math><mi>J</mi><mo>(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo>)</mo></math></span> for CW-pair <span><math><mi>A</mi><mover><mrow><mo>↪</mo></mrow><mrow><mi>i</mi></mrow></mover><mi>X</mi></math></span>, defined by B. Gray, which is homotopy equivalent to the homotopy fiber of the pinch map <span><math><mi>X</mi><msub><mrow><mo>∪</mo></mrow><mrow><mi>i</mi></mrow></msub><mi>C</mi><mi>A</mi><mo>→</mo><mi>Σ</mi><mi>A</mi></math></span>. As an application, we compute the 5 and 6-dim unstable homotopy groups of 3-dimensional mod <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup></math></span> Moore spaces for all positive integers <em>r</em>.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"356 ","pages":"Article 109043"},"PeriodicalIF":0.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142021245","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}
Pub Date : 2024-08-12DOI: 10.1016/j.topol.2024.109032
Yasushi Hirata, Yukinobu Yajima
This paper includes two main results. Dual discreteness is a well known generalization of D-spaces. The first one is that every Σ-product of compact metric spaces is dually discrete. The property aD is another generalization of D-spaces, and it implies irreducibility. The second one is that the product of many copies of is irreducible, where denotes an infinite countable discrete space.
本文包括两个主要结果。双离散性是 D 空间的一个众所周知的广义。第一个结果是紧凑度量空间的每个 Σ 积都是双离散的。aD 属性是 D 空间的另一种广义化，它意味着不可还原性。第二个是 N 的 ω1 多份的乘积 Nω1 是不可还原的，其中 N 表示无限可数离散空间。
{"title":"Dual discreteness of Σ-products and irreducibility of infinite products","authors":"Yasushi Hirata, Yukinobu Yajima","doi":"10.1016/j.topol.2024.109032","DOIUrl":"10.1016/j.topol.2024.109032","url":null,"abstract":"<div><p>This paper includes two main results. Dual discreteness is a well known generalization of <em>D</em>-spaces. The first one is that every <em>Σ</em>-product of compact metric spaces is dually discrete. The property <em>aD</em> is another generalization of <em>D</em>-spaces, and it implies irreducibility. The second one is that the product <span><math><msup><mrow><mi>N</mi></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup></math></span> of <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> many copies of <span><math><mi>N</mi></math></span> is irreducible, where <span><math><mi>N</mi></math></span> denotes an infinite countable discrete space.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"356 ","pages":"Article 109032"},"PeriodicalIF":0.6,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142021244","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}
Pub Date : 2024-08-08DOI: 10.1016/j.topol.2024.109031
Vlad Smolin
We prove that:
1.
If a Hausdorff M-space is a continuous closed image of a submetrizable space, then it is metrizable.
2.
A dense-in-itself open-closed image of a submetrizable space is submetrizable if and only if it is functionally Hausdorff and has a countable pseudocharacter.
3.
Let Y be a dense-in-itself space with the following property: . If Y is an open-closed image of a submetrizable space, then Y is submetrizable.
4.
There exist a submetrizable space X, a regular hereditarily paracompact non submetrizable first-countable space Y, and an open-closed map .
我们证明了这一点：
{"title":"Closed and open-closed images of submetrizable spaces","authors":"Vlad Smolin","doi":"10.1016/j.topol.2024.109031","DOIUrl":"10.1016/j.topol.2024.109031","url":null,"abstract":"<div><p>We prove that:</p><ul><li><span>1.</span><span><p>If a Hausdorff M-space is a continuous closed image of a submetrizable space, then it is metrizable.</p></span></li><li><span>2.</span><span><p>A dense-in-itself open-closed image of a submetrizable space is submetrizable if and only if it is functionally Hausdorff and has a countable pseudocharacter.</p></span></li><li><span>3.</span><span><p>Let <em>Y</em> be a dense-in-itself space with the following property: <span><math><mo>∀</mo><mi>y</mi><mo>∈</mo><mi>Y</mi><mspace></mspace><mo>∃</mo><mi>Q</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>⊆</mo><mi>Y</mi><mspace></mspace><mo>[</mo><mi>y</mi><mtext> is a non-isolated q-point in </mtext><mi>Q</mi><mo>(</mo><mi>y</mi><mo>)</mo><mo>]</mo></math></span>. If <em>Y</em> is an open-closed image of a submetrizable space, then <em>Y</em> is submetrizable.</p></span></li><li><span>4.</span><span><p>There exist a submetrizable space <em>X</em>, a regular hereditarily paracompact non submetrizable first-countable space <em>Y</em>, and an open-closed map <span><math><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></math></span>.</p></span></li></ul></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"356 ","pages":"Article 109031"},"PeriodicalIF":0.6,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141939797","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}
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