Topology and its Applications最新文献

{"title":"A complete invariant for shift equivalence for Boolean matrices and finite relations","authors":"","doi":"10.1016/j.topol.2024.109075","DOIUrl":"10.1016/j.topol.2024.109075","url":null,"abstract":"<div><div>We give a complete invariant for shift equivalence for Boolean matrices (equivalently finite relations), in terms of the period, the induced partial order on recurrent components, and the cohomology class of the relation on those components.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142417137","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":"On diagonal degrees and star networks","authors":"","doi":"10.1016/j.topol.2024.109074","DOIUrl":"10.1016/j.topol.2024.109074","url":null,"abstract":"<div><div>Given an open cover <span><math><mi>U</mi></math></span> of a topological space <em>X</em>, we introduce the notion of a star network for <span><math><mi>U</mi></math></span>. The associated cardinal function <span><math><mi>s</mi><mi>n</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>, where <span><math><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mi>s</mi><mi>n</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>, is used to establish new cardinal inequalities involving diagonal degrees. We show <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><mi>s</mi><mi>n</mi><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>Δ</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> for a <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> space <em>X</em>, giving a partial answer to a long-standing question of Angelo Bella. Many further results are given using variations of <span><math><mi>s</mi><mi>n</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>. One result has as corollaries Buzyakova's theorem that a ccc space with a regular <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>δ</mi></mrow></msub></math></span>-diagonal has cardinality at most <span><math><mi>c</mi></math></span>, as well as three results of Gotchev. Further results lead to logical improvements of theorems of Basile, Bella, and Ridderbos, a partial solution to a question of the same authors, and a theorem of Gotchev, Tkachenko, and Tkachuk. Finally, we define the Urysohn extent <span><math><mi>U</mi><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> with the property <span><math><mi>U</mi><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>≤</mo><mi>min</mi><mo>⁡</mo><mo>{</mo><mi>a</mi><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>,</mo><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>}</mo></math></span> and use the Erdős-Rado theorem to show that <span><math><mo>|</mo><mi>X</mi><mo>|</mo><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>U</mi><mi>e</mi><mo>(</mo><mi>X</mi><mo>)</mo><mover><mrow><mi>Δ</mi></mrow><mo>‾</mo></mover><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msup></math></span> for any Urysohn space <em>X</em>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142417071","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":"Approximations by disjoint subcontinua and indecomposability","authors":"","doi":"10.1016/j.topol.2024.109071","DOIUrl":"10.1016/j.topol.2024.109071","url":null,"abstract":"<div><div>We study approximations of continuum-wise connected spaces, or <em>semicontinua</em>, and show that every indecomposable semicontinuum can be approximated from within by a sequence of pairwise disjoint continua. As a corollary, we find that if <em>X</em> is a <em>G</em>-like continuum or a one-dimensional non-separating plane continuum, which is the closure of an indecomposable semicontinuum, then <em>X</em> is indecomposable. We also prove that a composant of an indecomposable continuum cannot be embedded into a Suslinian continuum.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142417070","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":"Calculation of Nielsen periodic numbers on infra-solvmanifolds","authors":"","doi":"10.1016/j.topol.2024.109073","DOIUrl":"10.1016/j.topol.2024.109073","url":null,"abstract":"<div><div>Recently, a formula for computing the Nielsen periodic numbers <span><math><mi>N</mi><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span> and <span><math><mi>N</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span> of self maps <em>f</em> on infra-nilmanifolds and infra-solvmanifolds of type (R) was found. In this paper, we extend this formula to the case of general infra-solvmanifolds. We show that infra-solvmanifolds are essentially reducible to the GCD and essentially toral, and determine conditions under which <span><math><mi>N</mi><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo><mo>=</mo><mi>N</mi><mo>(</mo><msup><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span>. We show that the prime Nielsen-Jiang periodic number <span><math><mi>N</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span> of a self map <em>f</em> on an infra-solvmanifold <em>M</em> can be calculated by Nielsen numbers of lifts of suitable iterates of <em>f</em> to an <span><math><mi>NR</mi></math></span>-solvmanifold that finitely covers <em>M</em>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142326718","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":"Hereditarily decomposable continua have non-block points","authors":"","doi":"10.1016/j.topol.2024.109072","DOIUrl":"10.1016/j.topol.2024.109072","url":null,"abstract":"<div><div>In this note we expand upon our results from <span><span>[1]</span></span> to show that every nondegenerate hereditarily decomposable Hausdorff continuum has two or more non-block points, i.e. points whose complements contain a continuum-connected dense subset. The celebrated non-cut point existence theorem states that all nondegenerate Hausdorff continua have two or more non-cut points, and the corresponding result for non-block points is known to hold for metrizable continua. It is also known that there are consistent examples of Hausdorff continua with no non-block points, but that non-block point existence holds for Hausdorff continua that are either aposyndetic, irreducible, or separable.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250245","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":"The Jones polynomial for a torus knot with twists","authors":"","doi":"10.1016/j.topol.2024.109069","DOIUrl":"10.1016/j.topol.2024.109069","url":null,"abstract":"<div><p>We compute the Jones polynomial for a three-parameter family of links, the twisted torus links of the form <span><math><mi>T</mi><mo>(</mo><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo><mo>,</mo><mo>(</mo><mn>2</mn><mo>,</mo><mi>s</mi><mo>)</mo><mo>)</mo></math></span> where <em>p</em> and <em>q</em> are coprime and <em>s</em> is nonzero. When <span><math><mi>s</mi><mo>=</mo><mn>2</mn><mi>n</mi></math></span>, these links are the twisted torus knots <span><math><mi>T</mi><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>;</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span>. We show that for <span><math><mi>T</mi><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>;</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span>, the Jones polynomial is trivial if and only if the knot is trivial.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142242462","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":"More on Whitney levels of some decomposable continua","authors":"","doi":"10.1016/j.topol.2024.109068","DOIUrl":"10.1016/j.topol.2024.109068","url":null,"abstract":"<div><p>In this paper, we show that there exists a non-<em>D</em>-continuum such that each positive Whitney level of the hyperspace of subcontinua of the continuum is both <span><math><msup><mrow><mi>D</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> and Wilder. We show that the property of being continuum-wise Wilder is not a Whitney property, while it is a Whitney reversible property. Furthermore, we introduce the new class of continua: closed set-wise Wilder continua. This class is larger than the class of continuum chainable continua and smaller than the class of continuum-wise Wilder continua. In addition to the above results, we show that the Cartesian product of two closed set-wise Wilder continua is close set-wise Wilder.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142242463","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":"One-point connectifications of regular spaces","authors":"","doi":"10.1016/j.topol.2024.109060","DOIUrl":"10.1016/j.topol.2024.109060","url":null,"abstract":"<div><p>It is well known that, a locally compact Hausdorff space has a Hausdorff one-point compactification (known as the <em>Alexandroff compactification</em>) if and only if it is non-compact. There is also, an old question of Alexandroff of characterizing spaces which have a one-point connectification. Here, we study one-point connectifications in the realm of regular spaces and prove that a locally connected space has a regular one-point connectification if and only if the space has no regular-closed component. This, also gives an answer to the conjecture raised by M. R. Koushesh. Then, we consider the set of all one-point connectifications of a locally connected regular space and show that, this set (naturally partially ordered) is a compact conditionally complete lattice. Further, we extend our theorem for locally connected regular spaces with a topological property <span><math><mi>P</mi></math></span> and give conditions on <span><math><mi>P</mi></math></span> which guarantee the space to have a regular one-point connectification with <span><math><mi>P</mi></math></span>.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204842","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":"The Macías topology on integral domains","authors":"","doi":"10.1016/j.topol.2024.109070","DOIUrl":"10.1016/j.topol.2024.109070","url":null,"abstract":"<div><p>In this manuscript a recent topology on the positive integers generated by the collection of <span><math><mo>{</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>}</mo></math></span> where <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mo>=</mo><mo>{</mo><mi>m</mi><mo>:</mo><mi>gcd</mi><mo>⁡</mo><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>}</mo></math></span> is generalized over integral domains. Some of its topological properties are studied. Properties of this topology on infinite principal ideal domains that are not fields are also explored, and a new topological proof of the infinitude of prime elements is obtained (assuming the set of units is finite or not open), different from those presented in the style of H. Furstenberg. Finally, some problems are proposed.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142173849","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 family of slice-torus invariants from the divisibility of Lee classes","authors":"","doi":"10.1016/j.topol.2024.109059","DOIUrl":"10.1016/j.topol.2024.109059","url":null,"abstract":"<div><p>We give a family of slice-torus invariants <span><math><msub><mrow><mover><mrow><mi>s</mi><mi>s</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>c</mi></mrow></msub></math></span>, each defined from the <em>c</em>-divisibility of the reduced Lee class in a variant of reduced Khovanov homology, parameterized by prime elements <em>c</em> in any principal ideal domain <em>R</em>. For the special case <span><math><mo>(</mo><mi>R</mi><mo>,</mo><mi>c</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>F</mi><mo>[</mo><mi>H</mi><mo>]</mo><mo>,</mo><mi>H</mi><mo>)</mo></math></span> where <em>F</em> is any field, we prove that <span><math><msub><mrow><mover><mrow><mi>s</mi><mi>s</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>c</mi></mrow></msub></math></span> coincides with the Rasmussen invariant <span><math><msup><mrow><mi>s</mi></mrow><mrow><mi>F</mi></mrow></msup></math></span> over <em>F</em>. Compared with the unreduced invariants <span><math><mi>s</mi><msub><mrow><mi>s</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> defined by the first author in a previous paper, we prove that <span><math><mi>s</mi><msub><mrow><mi>s</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><msub><mrow><mover><mrow><mi>s</mi><mi>s</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>c</mi></mrow></msub></math></span> for <span><math><mo>(</mo><mi>R</mi><mo>,</mo><mi>c</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>F</mi><mo>[</mo><mi>H</mi><mo>]</mo><mo>,</mo><mi>H</mi><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>Z</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. However for <span><math><mo>(</mo><mi>R</mi><mo>,</mo><mi>c</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>Z</mi><mo>,</mo><mn>3</mn><mo>)</mo></math></span>, computational results show that <span><math><mi>s</mi><msub><mrow><mi>s</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> is not slice-torus, which implies that it is linearly independent from the reduced invariants, and particularly from the Rasmussen invariants.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204846","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}