Topology and its Applications最新文献

{"title":"A proof of the generalized geometric boundary theorem using filtered spectra","authors":"Sihao Ma","doi":"10.1016/j.topol.2024.109006","DOIUrl":"10.1016/j.topol.2024.109006","url":null,"abstract":"<div><p>In <span>[1, Lem. A.4.1]</span>, Behrens generalized the classical geometric boundary theorem <span>[10, Thm. 2.3.4]</span>. In this article, we will reformulate <span>[1, Lem. A.4.1]</span> to fix a mistake made by Behrens, and prove it using the language of filtered spectra.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"355 ","pages":"Article 109006"},"PeriodicalIF":0.6,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141546679","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}

{"title":"Every finite-dimensional analytic space is σ-homogeneous","authors":"Claudio Agostini,&nbsp;Andrea Medini","doi":"10.1016/j.topol.2024.109004","DOIUrl":"https://doi.org/10.1016/j.topol.2024.109004","url":null,"abstract":"<div><p>All spaces are assumed to be separable and metrizable. Building on work of van Engelen, Harrington, Michalewski and Ostrovsky, we obtain the following results:</p><ul><li><span>•</span><span><p>Every finite-dimensional analytic space is <em>σ</em>-homogeneous with analytic witnesses,</p></span></li><li><span>•</span><span><p>Every finite-dimensional analytic space is <em>σ</em>-homogeneous with pairwise disjoint <span><math><msubsup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> witnesses.</p></span></li></ul> Furthermore, the complexity of the witnesses is optimal in both of the above results. This answers a question of Medini and Vidnyánszky, and completes the picture regarding <em>σ</em>-homogeneity in the finite-dimensional realm. It is an open problem whether every analytic space is <em>σ</em>-homogeneous. We also investigate finite unions of homogeneous spaces.</div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"355 ","pages":"Article 109004"},"PeriodicalIF":0.6,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0166864124001895/pdfft?md5=f80eb0bcf017e7b3b591f58517190ed2&pid=1-s2.0-S0166864124001895-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141605037","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}

{"title":"Locally homeomorphic infinite Lindelof P-groups are homeomorphic","authors":"Mikhail Tkachenko","doi":"10.1016/j.topol.2024.109005","DOIUrl":"10.1016/j.topol.2024.109005","url":null,"abstract":"<div><p>We prove the statement formulated in the title of the article. Then we apply it to show that there exists Lindelöf <em>P</em>-groups <em>G</em> and <em>H</em> satisfying <span><math><mi>w</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>w</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>=</mo><mo>|</mo><mi>G</mi><mo>|</mo><mo>=</mo><mo>|</mo><mi>H</mi><mo>|</mo><mo>=</mo><msub><mrow><mi>ℵ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> such that <em>G</em> and <em>H</em> are not locally homeomorphic. This solves Problem 4.4.7 from the book (Arhangel'skii and Tkachenko, 2008 <span>[1]</span>) in the negative. Also, we present two homeomorphic complete Abelian <em>P</em>-groups one of which is <em>ω</em>-narrow and the other is not.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"355 ","pages":"Article 109005"},"PeriodicalIF":0.6,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552347","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}

{"title":"Estimates on the topological Hausdorff dimensions of fractal squares","authors":"Jian-Ci Xiao","doi":"10.1016/j.topol.2024.109003","DOIUrl":"10.1016/j.topol.2024.109003","url":null,"abstract":"<div><p>We first obtain some upper bounds on the topological Hausdorff dimensions of fractal squares. As a corollary, we give the formula for this dimension of a special class of fractal squares. Combined with previous results, we also complete the computation of the topological Hausdorff dimensions of fractal squares of order three. Some of these require non-trivial constructions of bases. Our results also shed light on the Lipschitz classification of fractal squares.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"355 ","pages":"Article 109003"},"PeriodicalIF":0.6,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552348","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}

{"title":"Corrigendum to “Typical dynamics of Newton's method” [Topol. Appl. 318 (2022) 108201]","authors":"Jan Dudák ,&nbsp;T.H. Steele","doi":"10.1016/j.topol.2024.108986","DOIUrl":"https://doi.org/10.1016/j.topol.2024.108986","url":null,"abstract":"<div><p>Let <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo></math></span> be the space of continuously differentiable real-valued functions defined on <span><math><mo>[</mo><mo>−</mo><mi>M</mi><mo>,</mo><mi>M</mi><mo>]</mo></math></span>. Here, we address an irremediable flaw found in <span>[4]</span>, and show that for the typical element <em>f</em> in <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo></math></span>, there exists a set <span><math><mi>S</mi><mo>⊆</mo><mo>[</mo><mo>−</mo><mi>M</mi><mo>,</mo><mi>M</mi><mo>]</mo></math></span>, both residual and of full measure in <span><math><mo>[</mo><mo>−</mo><mi>M</mi><mo>,</mo><mi>M</mi><mo>]</mo></math></span>, such that for any <span><math><mi>x</mi><mo>∈</mo><mi>S</mi></math></span>, the trajectory generated by Newton's method using <em>f</em> and <em>x</em> either diverges, converges to a root of <em>f</em>, or generates a Cantor set as its attractor. Whenever the Cantor set is the attractor, the dynamics on the attractor are described by a single type of adding machine, so that the dynamics on all of these attracting Cantor sets are topologically equivalent.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"354 ","pages":"Article 108986"},"PeriodicalIF":0.6,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0166864124001718/pdfft?md5=cee850ac49fa85977ac1cc21ab194a29&pid=1-s2.0-S0166864124001718-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141481107","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}

{"title":"A cork of the rational surface with the second Betti number 9","authors":"Yohei Wakamaki","doi":"10.1016/j.topol.2024.109002","DOIUrl":"https://doi.org/10.1016/j.topol.2024.109002","url":null,"abstract":"<div><p>We provide the first explicit example of a cork of <span><math><msup><mrow><mi>CP</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>#</mi><mn>8</mn><mover><mrow><msup><mrow><mi>CP</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mo>‾</mo></mover></math></span>. This result gives the current smallest second Betti number of a standard simply-connected closed 4-manifold for which an explicit cork has been found.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"354 ","pages":"Article 109002"},"PeriodicalIF":0.6,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141481109","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}

{"title":"Artin presentations of the trivial group and hyperbolic closed pure 3-braids","authors":"Lorena Armas-Sanabria ,&nbsp;Jesús Rodríguez Viorato ,&nbsp;E. Fanny Jasso-Hernández","doi":"10.1016/j.topol.2024.108989","DOIUrl":"10.1016/j.topol.2024.108989","url":null,"abstract":"<div><p>We consider a special class of framed links that arise from the hexatangle. Such links are introduced in <span>[3]</span>, where it was also analyzed when the 3-manifold obtained after performing integral Dehn surgery on closed pure 3-braids is <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. In the present paper, we analyze the symmetries of the hexatangle and give a list of Artin <em>n</em>-presentations for the trivial group. These presentations correspond to the double-branched covers of the hexatangle that produce <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> after Dehn surgery. Also, using a result of Birman and Menasco <span>[4]</span>, we determine which closed pure 3-braids are hyperbolic.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"354 ","pages":"Article 108989"},"PeriodicalIF":0.6,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141586129","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}

{"title":"A short elementary proof of Beben and Theriault's theorem on homotopy fibers","authors":"Daisuke Kishimoto ,&nbsp;Yuki Minowa","doi":"10.1016/j.topol.2024.108998","DOIUrl":"10.1016/j.topol.2024.108998","url":null,"abstract":"<div><p>Beben and Theriault proved a theorem on the homotopy fiber of an extension of a map with respect to a cone attachment, which has produced several applications. We give a short and elementary proof of this theorem.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"355 ","pages":"Article 108998"},"PeriodicalIF":0.6,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141586133","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}

{"title":"Point-set games and functions with the hereditary small oscillation property","authors":"Marek Balcerzak ,&nbsp;Tomasz Natkaniec ,&nbsp;Piotr Szuca","doi":"10.1016/j.topol.2024.109000","DOIUrl":"https://doi.org/10.1016/j.topol.2024.109000","url":null,"abstract":"<div><p>Given a metric space <em>X</em>, we consider certain families of functions <span><math><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>R</mi></math></span> having the hereditary oscillation property HSOP and the hereditary continuous restriction property HCRP on large sets. When <em>X</em> is Polish, among them there are families of Baire measurable functions, <span><math><mover><mrow><mi>μ</mi></mrow><mo>‾</mo></mover></math></span>-measurable functions (for a finite nonatomic Borel measure <em>μ</em> on <em>X</em>) and Marczewski measurable functions. We obtain their characterizations using a class of equivalent point-set games. In similar aspects, we study cliquish functions, SZ-functions and countably continuous functions.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"354 ","pages":"Article 109000"},"PeriodicalIF":0.6,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0166864124001858/pdfft?md5=e79a82c13d4f51b41f2558f42380e6dd&pid=1-s2.0-S0166864124001858-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141481143","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}

{"title":"On some kinds of ω-balancedness and (*) properties in certain semitopological groups","authors":"Liang-Xue Peng","doi":"10.1016/j.topol.2024.109001","DOIUrl":"https://doi.org/10.1016/j.topol.2024.109001","url":null,"abstract":"<div><p>In this article, we discuss some relationships of <em>ω</em>-balancedness and <span><math><mo>(</mo><mo>⁎</mo><mo>)</mo></math></span> properties which were introduced for giving characterizations of subgroups of topological products of certain para(semi)topological groups. We mainly get the following results.</p><p>If <em>G</em> is a regular <em>ω</em>-balanced locally <em>ω</em>-good semitopological group with a <em>q</em>-point, then <span><math><mi>I</mi><mi>r</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span> if and only if <span><math><mi>S</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span>. If <em>G</em> is a regular strongly paracompact semitopological group with a <em>q</em>-point and <span><math><mi>S</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span>, then <em>G</em> is completely <em>ω</em>-balanced if and only if <em>G</em> has property <span><math><mo>(</mo><msup><mrow></mrow><mrow><mo>⁎</mo></mrow></msup><mo>)</mo></math></span>. If <em>G</em> is a regular paracompact <em>ω</em>-balanced locally good semitopological group with a <em>q</em>-point and <span><math><mi>S</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span>, then <em>G</em> has property <span><math><mo>(</mo><mi>w</mi><mo>⁎</mo><mo>)</mo></math></span> if and only if <em>G</em> has property (**). If <em>G</em> is a regular metacompact semitopological group with a <em>q</em>-point and <span><math><mi>S</mi><mi>m</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>ω</mi></math></span>, then <em>G</em> is <em>MM</em>-<em>ω</em>-balanced if and only if <em>G</em> is <em>M</em>-<em>ω</em>-balanced.</p><p>We show that a semitopological group <em>G</em> admits a homeomorphic embedding as a subgroup of a product of metrizable semitopological groups if and only if <em>G</em> is topologically isomorphic to a subgroup of a product of semitopological groups which are first-countable paracompact regular <em>σ</em>-spaces and is topologically isomorphic to a subgroup of a product of Moore semitopological groups.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"354 ","pages":"Article 109001"},"PeriodicalIF":0.6,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141481108","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}