Pub Date : 2024-10-02DOI: 10.1016/j.topol.2024.109076
Alexander V. Osipov
A space X is sequentially separable if there is a countable such that every point of X is the limit of a sequence of points from S. In 2004, N.V. Velichko defined and investigated concepts close to sequential separability: σ-separability and F-separability. The aim of this paper is to study σ-separability and F-separability (and their hereditary variants) of the space of all real-valued continuous functions, defined on a Tychonoff space X, endowed with the pointwise convergence topology. In particular, we proved that σ-separability coincides with sequential separability. Hereditary variants (hereditarily σ-separability and hereditarily F-separability) coincide with Fréchet–Urysohn property in the class of cosmic spaces.
如果存在一个可数 S⊂X，使得 X 的每个点都是来自 S 的点序列的极限，则空间 X 是顺序可分的。2004 年，N.V. Velichko 定义并研究了与顺序可分性接近的概念：σ 可分性和 F 可分性。本文的目的是研究所有实值连续函数空间 Cp(X) 的 σ 可分性和 F 可分性（及其遗传变异），这些函数定义在泰克诺夫空间 X 上，并赋有点收敛拓扑。我们特别证明了 σ 可分性与顺序可分性重合。在宇宙空间类中，遗传变异（遗传σ可分性和遗传F可分性）与弗雷谢特-乌里索恩性质重合。
{"title":"Velichko's notions close to sequential separability and their hereditary variants in Cp-theory","authors":"Alexander V. Osipov","doi":"10.1016/j.topol.2024.109076","DOIUrl":"10.1016/j.topol.2024.109076","url":null,"abstract":"<div><div>A space <em>X</em> is <em>sequentially separable</em> if there is a countable <span><math><mi>S</mi><mo>⊂</mo><mi>X</mi></math></span> such that every point of <em>X</em> is the limit of a sequence of points from <em>S</em>. In 2004, N.V. Velichko defined and investigated concepts close to sequential separability: <em>σ-separability</em> and <em>F-separability</em>. The aim of this paper is to study <em>σ</em>-separability and <em>F</em>-separability (and their hereditary variants) of the space <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> of all real-valued continuous functions, defined on a Tychonoff space <em>X</em>, endowed with the pointwise convergence topology. In particular, we proved that <em>σ</em>-separability coincides with sequential separability. Hereditary variants (hereditarily <em>σ</em>-separability and hereditarily <em>F</em>-separability) coincide with Fréchet–Urysohn property in the class of cosmic spaces.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":"357 ","pages":"Article 109076"},"PeriodicalIF":0.6,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142417072","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 : 2024-10-02DOI: 10.1016/j.topol.2024.109075
Ethan Akin , Marian Mrozek , Mateusz Przybylski , Jim Wiseman
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.
{"title":"A complete invariant for shift equivalence for Boolean matrices and finite relations","authors":"Ethan Akin , Marian Mrozek , Mateusz Przybylski , Jim Wiseman","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":"357 ","pages":"Article 109075"},"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}
Pub Date : 2024-10-01DOI: 10.1016/j.topol.2024.109074
Nathan Carlson
Given an open cover of a topological space X, we introduce the notion of a star network for . The associated cardinal function , where , is used to establish new cardinal inequalities involving diagonal degrees. We show for a space X, giving a partial answer to a long-standing question of Angelo Bella. Many further results are given using variations of . One result has as corollaries Buzyakova's theorem that a ccc space with a regular -diagonal has cardinality at most , 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 with the property and use the Erdős-Rado theorem to show that for any Urysohn space X.
给定拓扑空间 X 的开盖 U，我们引入 U 的星形网络概念。相关的心形函数 sn(X)（其中 e(X)≤sn(X)≤L(X) ）用于建立涉及对角度的新心形不等式。我们证明了 T1 空间 X 的 |X|≤sn(X)Δ(X)，部分回答了安杰洛-贝拉（Angelo Bella）的一个长期问题。利用 sn(X) 的变化给出了许多进一步的结果。其中一个结果的推论是布扎科娃（Buzyakova）的定理，即具有规则 Gδ 对角线的ccc 空间的心性至多为 c，以及哥切夫（Gotchev）的三个结果。进一步的结果导致了巴西尔、贝拉和里德博斯定理的逻辑改进，同一作者的一个问题的部分解答，以及哥切夫、特卡琴科和特卡丘克的一个定理。最后，我们定义了具有 Ue(X)≤min{aL(X),e(X)} 特性的乌里索恩程度 Ue(X)，并使用厄尔多斯-拉多定理证明了对于任何乌里索恩空间 X，|X|≤2Ue(X)Δ‾(X)。
{"title":"On diagonal degrees and star networks","authors":"Nathan Carlson","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":"357 ","pages":"Article 109074"},"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}
Pub Date : 2024-09-27DOI: 10.1016/j.topol.2024.109071
David S. Lipham
We study approximations of continuum-wise connected spaces, or semicontinua, 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 X is a G-like continuum or a one-dimensional non-separating plane continuum, which is the closure of an indecomposable semicontinuum, then X is indecomposable. We also prove that a composant of an indecomposable continuum cannot be embedded into a Suslinian continuum.
我们研究了连续面连接空间或半连续面的近似，并证明每个不可分解的半连续面都可以从内部被一连串成对相离的连续面近似。作为推论，我们发现如果 X 是类 G 连续体或一维非分离平面连续体（即不可分解半连续体的闭包），那么 X 是不可分解的。我们还证明了不可分解连续统的合成子不能嵌入到苏斯林连续统中。
{"title":"Approximations by disjoint subcontinua and indecomposability","authors":"David S. Lipham","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":"357 ","pages":"Article 109071"},"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}
Pub Date : 2024-09-19DOI: 10.1016/j.topol.2024.109073
Changbok Li
Recently, a formula for computing the Nielsen periodic numbers and of self maps f 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 . We show that the prime Nielsen-Jiang periodic number of a self map f on an infra-solvmanifold M can be calculated by Nielsen numbers of lifts of suitable iterates of f to an -solvmanifold that finitely covers M.
最近，我们发现了一个公式，用于计算下零曼形和下溶曼形上自映射 f 的尼尔森周期数 NFn(f) 和 NPn(f)。在本文中，我们将这一公式推广到一般的下溶点。我们证明了下溶漫游体本质上可还原为 GCD，本质上是环状的，并确定了 NFn(f)=N(fn) 的条件。我们证明了下溶漫性 M 上自映射 f 的素数尼尔森-蒋周期数 NPn(f) 可以通过 f 向有限覆盖 M 的 NR 溶漫性的合适迭代的提升的尼尔森数来计算。
{"title":"Calculation of Nielsen periodic numbers on infra-solvmanifolds","authors":"Changbok Li","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":"357 ","pages":"Article 109073"},"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}
Pub Date : 2024-09-16DOI: 10.1016/j.topol.2024.109072
Daron Anderson
In this note we expand upon our results from [1] 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.
{"title":"Hereditarily decomposable continua have non-block points","authors":"Daron Anderson","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":"357 ","pages":"Article 109072"},"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}
Pub Date : 2024-09-11DOI: 10.1016/j.topol.2024.109069
Brandon Bavier , Brandy Doleshal
We compute the Jones polynomial for a three-parameter family of links, the twisted torus links of the form where p and q are coprime and s is nonzero. When , these links are the twisted torus knots . We show that for , the Jones polynomial is trivial if and only if the knot is trivial.
{"title":"The Jones polynomial for a torus knot with twists","authors":"Brandon Bavier , Brandy Doleshal","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":"357 ","pages":"Article 109069"},"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}
In this paper, we show that there exists a non-D-continuum such that each positive Whitney level of the hyperspace of subcontinua of the continuum is both 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.
在本文中，我们证明存在一个非 D 连续统，使得连续统子连续统超空间的每个正惠特尼级既是 D⁎ 又是 Wilder。我们证明，连续度 Wilder 的性质不是惠特尼性质，而它是惠特尼可逆性质。此外，我们还引入了一类新的连续体：封闭集智 Wilder 连续体。这一类连续体比可链连续体大，比连续体-明智怀尔德连续体小。除了上述结果，我们还证明了两个闭集智怀尔德连续体的笛卡儿积是闭集智怀尔德连续体。
{"title":"More on Whitney levels of some decomposable continua","authors":"Alejandro Illanes , Eiichi Matsuhashi , Yoshiyuki Oshima","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":"357 ","pages":"Article 109068"},"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}
Pub Date : 2024-09-10DOI: 10.1016/j.topol.2024.109060
Nidaa Hasan Haji , Abdolaziz Hesari , Rafid Habib Buti
It is well known that, a locally compact Hausdorff space has a Hausdorff one-point compactification (known as the Alexandroff compactification) 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 and give conditions on which guarantee the space to have a regular one-point connectification with .
众所周知，当且仅当局部紧凑的豪斯多夫空间是非紧凑时，它才具有豪斯多夫一点紧凑化（称为）。此外，亚历山德罗夫还提出了一个老问题，即如何描述具有单点连接的空间。在这里，我们研究正则空间领域中的一点连通，并证明当且仅当局部连通空间没有正则封闭成分时，该空间才具有正则一点连通。这也解答了 M. R. Koushesh 提出的猜想。然后，我们考虑了局部连通正则空间的所有单点连通的集合，并证明这个集合（自然部分有序）是一个紧凑的条件完全网格。此外，我们还扩展了具有拓扑性质的局部相连正则空间的定理，并给出了保证空间具有......的正则单点连接的条件。
{"title":"One-point connectifications of regular spaces","authors":"Nidaa Hasan Haji , Abdolaziz Hesari , Rafid Habib Buti","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":"357 ","pages":"Article 109060"},"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}
Pub Date : 2024-09-10DOI: 10.1016/j.topol.2024.109070
Jhixon Macías
In this manuscript a recent topology on the positive integers generated by the collection of where 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.
{"title":"The Macías topology on integral domains","authors":"Jhixon Macías","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":"357 ","pages":"Article 109070"},"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}
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