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The Macias topology on integral domains 积分域上的马西亚斯拓扑学
arXiv - MATH - General Topology Pub Date : 2024-06-07 DOI: arxiv-2406.04623
Jhixon Macías
{"title":"The Macias topology on integral domains","authors":"Jhixon Macías","doi":"arxiv-2406.04623","DOIUrl":"https://doi.org/arxiv-2406.04623","url":null,"abstract":"In this manuscript, a recent topology on the positive integers, generated by\u0000the collection of relatively prime positive integers, is generalized over\u0000integral domains. Some of its topological properties are studied. Properties of\u0000this topology on infinite principal ideal domains that are not fields are also\u0000explored, and a new topological proof of the infinitude of prime elements is\u0000obtained (assuming the set of units is finite), different from those presented\u0000in the style of H. Furstenberg. Finally, some problems are proposed.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fixed point results for contractions of polynomial type 多项式类型收缩的定点结果
arXiv - MATH - General Topology Pub Date : 2024-06-05 DOI: arxiv-2406.03446
Mohamed Jleli, Cristina Maria Pacurar, Bessem Samet
{"title":"Fixed point results for contractions of polynomial type","authors":"Mohamed Jleli, Cristina Maria Pacurar, Bessem Samet","doi":"arxiv-2406.03446","DOIUrl":"https://doi.org/arxiv-2406.03446","url":null,"abstract":"We introduce two new classes of single-valued contractions of polynomial type\u0000defined on a metric space. For the first one, called the class of polynomial\u0000contractions, we establish two fixed point theorems. Namely, we first consider\u0000the case when the mapping is continuous. Next, we weaken the continuity\u0000condition. In particular, we recover Banach's fixed point theorem. The second\u0000class, called the class of almost polynomial contractions, includes the class\u0000of almost contractions introduced by Berinde [Nonlinear Analysis Forum. 9(1)\u0000(2004) 43--53]. A fixed point theorem is established for almost polynomial\u0000contractions. The obtained result generalizes that derived by Berinde in the\u0000above reference. Several examples showing that our generalizations are\u0000significant, are provided.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141524549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Velichko's notions close to sequentially separability and their hereditary variants in $C_p$-theory 韦利奇科接近于顺序可分性的概念及其在$C_p$理论中的遗传变异
arXiv - MATH - General Topology Pub Date : 2024-06-05 DOI: arxiv-2406.03014
Alexander V. Osipov
{"title":"Velichko's notions close to sequentially separability and their hereditary variants in $C_p$-theory","authors":"Alexander V. Osipov","doi":"arxiv-2406.03014","DOIUrl":"https://doi.org/arxiv-2406.03014","url":null,"abstract":"A space $X$ is sequentially separable if there is a countable $Ssubset X$\u0000such that every point of $X$ is the limit of a sequence of points from $S$. In\u00002004, N.V. Velichko defined and investigated concepts close to sequentially\u0000separable: $sigma$-separability and $F$-separability. The aim of this paper is\u0000to study $sigma$-separability and $F$-separability (and their hereditary\u0000variants) of the space $C_p(X)$ of all real-valued continuous functions,\u0000defined on a Tychonoff space $X$, endowed with the pointwise convergence\u0000topology. In particular, we proved that $sigma$-separability coincides with\u0000sequential separability. Hereditary variants (hereditarily $sigma$-separablity\u0000and hereditarily $F$-separablity) coincides with Frechet-Urysohn property in\u0000the class of cosmic spaces.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141524550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Borromean Hypergraph Formation in Dense Random Rectangles 密集随机矩形中的博罗姆超图形成
arXiv - MATH - General Topology Pub Date : 2024-05-31 DOI: arxiv-2405.20874
Alexander R. Klotz
{"title":"Borromean Hypergraph Formation in Dense Random Rectangles","authors":"Alexander R. Klotz","doi":"arxiv-2405.20874","DOIUrl":"https://doi.org/arxiv-2405.20874","url":null,"abstract":"We develop a minimal system to study the stochastic formation of Borromean\u0000links within topologically entangled networks without requiring the use of knot\u0000invariants. Borromean linkages may form in entangled solutions of open polymer\u0000chains or in Olympic gel systems such as kinetoplast DNA, but it is challenging\u0000to investigate this due to the difficulty of computing three-body link\u0000invariants. Here, we investigate randomly oriented rectangles densely packed\u0000within a volume, and evaluate them for Hopf linking and Borromean link\u0000formation. We show that dense packings of rectangles can form Borromean\u0000triplets and larger clusters, and that in high enough density the combination\u0000of Hopf and Borromean linking can create a percolating hypergraph through the\u0000network. We present data for the percolation threshold of Borromean\u0000hypergraphs, and discuss implications for the existence of Borromean\u0000connectivity within kinetoplast DNA.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141259547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Graphs and Their Vietoris-Rips Complexes Have the Same Pseudotopological Weak Homotopy Type 图形及其维托里斯-里普斯复合物具有相同的伪拓扑弱同调类型
arXiv - MATH - General Topology Pub Date : 2024-05-31 DOI: arxiv-2406.00149
Jonathan Treviño-Marroquín
{"title":"Graphs and Their Vietoris-Rips Complexes Have the Same Pseudotopological Weak Homotopy Type","authors":"Jonathan Treviño-Marroquín","doi":"arxiv-2406.00149","DOIUrl":"https://doi.org/arxiv-2406.00149","url":null,"abstract":"In this document, we propose a bridge between the graphs and the geometric\u0000realizations of their Vietoris Rips complexes, i.e. Graphs, with their\u0000canonical v{C}ech closure structure, have the same homotopy type that the\u0000realization of their Vietoris Rips complex.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141258898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Henderson-like interval maps 类似亨德森区间图
arXiv - MATH - General Topology Pub Date : 2024-05-30 DOI: arxiv-2405.20533
Jernej Činč
{"title":"Henderson-like interval maps","authors":"Jernej Činč","doi":"arxiv-2405.20533","DOIUrl":"https://doi.org/arxiv-2405.20533","url":null,"abstract":"In this paper we study interval maps with zero topological entropy that are\u0000crooked; i.e. whose inverse limit is the pseudo-arc. We show that there are\u0000uncountably many pairwise non-conjugate zero entropy crooked interval maps with\u0000different sets of fixed points. We also show that there are uncountably many\u0000crooked maps that are pairwise non-conjugate and have exactly two fixed points.\u0000Furthermore, we provide a characterization of crooked interval maps that are\u0000under (above) the diagonal.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"194 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141258803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Lattices of slowly oscillating functions 缓慢振荡函数的网格
arXiv - MATH - General Topology Pub Date : 2024-05-29 DOI: arxiv-2405.19555
Yutaka Iwamoto
{"title":"Lattices of slowly oscillating functions","authors":"Yutaka Iwamoto","doi":"arxiv-2405.19555","DOIUrl":"https://doi.org/arxiv-2405.19555","url":null,"abstract":"We show that lattice isomorphisms between the lattices of slowly oscillating\u0000functions on chain-connected proper metric spaces induce coarsely equivalent\u0000homeomorphisms. This result leads to a Banach-Stone-type theorem for these\u0000lattices. Furthermore, we provide a representation theorem that characterizes\u0000linear lattice isomorphisms among these lattices.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"87 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141195482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Dedekind-MacNeille and related completions: subfitness, regularity, and Booleanness 戴德金-麦克尼尔和相关补全：次拟合、正则性和布尔性
arXiv - MATH - General Topology Pub Date : 2024-05-29 DOI: arxiv-2405.19171
G. Bezhanishvili, F. Dashiell Jr, A. Moshier, J. Walters-Wayland
{"title":"Dedekind-MacNeille and related completions: subfitness, regularity, and Booleanness","authors":"G. Bezhanishvili, F. Dashiell Jr, A. Moshier, J. Walters-Wayland","doi":"arxiv-2405.19171","DOIUrl":"https://doi.org/arxiv-2405.19171","url":null,"abstract":"Completions play an important r^ole for studying structure by supplying\u0000elements that in some sense ``ought to be.\" Among these, the Dedekind-MacNeille\u0000completion is of particular importance. In 1968 Janowitz provided necessary and\u0000sufficient conditions for it to be subfit or Boolean. Another natural\u0000separation axiom situated between the two is regularity. We explore similar\u0000characterizations of when closely related completions are subfit, regular, or\u0000Boolean. We are mainly interested in the Bruns-Lakser, ideal, and canonical\u0000completions, which are useful in pointfree topology since (unlike the\u0000Dedekind-MacNeille completion) they satisfy stronger forms of distributivity.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141195428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
All iterated function systems are Lipschitz up to an equivalent metric 所有迭代函数系统都是 Lipschitz 的，直到等价度量为止
arXiv - MATH - General Topology Pub Date : 2024-05-27 DOI: arxiv-2405.16977
Michał Popławski
{"title":"All iterated function systems are Lipschitz up to an equivalent metric","authors":"Michał Popławski","doi":"arxiv-2405.16977","DOIUrl":"https://doi.org/arxiv-2405.16977","url":null,"abstract":"A finite family $mathcal{F}={f_1,ldots,f_n}$ of continuous selfmaps of a\u0000given metric space $X$ is called an iterated function system (shortly IFS). In\u0000a case of contractive selfmaps of a complete metric space is well-known that\u0000IFS has an unique attractor cite{Hu}. However, in cite{LS} authors studied\u0000highly non-contractive IFSs, i.e. such families\u0000$mathcal{F}={f_1,ldots,f_n}$ of continuous selfmaps that for any\u0000remetrization of $X$ each function $f_i$ has Lipschitz constant $>1,\u0000i=1,ldots,n.$ They asked when one can remetrize $X$ that $mathcal{F}$ is\u0000Lipschitz IFS, i.e. all $f_i's$ are Lipschitz (not necessarily contractive), $\u0000i=1,ldots,n$. We give a general positive answer for this problem by\u0000constructing respective new metric (equivalent to the original one) on $X$,\u0000determined by a given family $mathcal{F}={f_1,ldots,f_n}$ of continuous\u0000selfmaps of $X$. However, our construction is valid even for some specific\u0000infinite families of continuous functions.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141168543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Invertibility in nonassociative ordered rings and in weak-quasi-topological nonassociative rings 非关联有序环和弱准拓扑非关联环中的不可逆性
arXiv - MATH - General Topology Pub Date : 2024-05-26 DOI: arxiv-2405.16565
Nizar El Idrissi, Hicham Zoubeir
{"title":"Invertibility in nonassociative ordered rings and in weak-quasi-topological nonassociative rings","authors":"Nizar El Idrissi, Hicham Zoubeir","doi":"arxiv-2405.16565","DOIUrl":"https://doi.org/arxiv-2405.16565","url":null,"abstract":"Invertibility is important in ring theory because it enables division and\u0000facilitates solving equations. Moreover, rings can be endowed with extra\u0000''structure'' such as order and topology that add new properties. The two main\u0000theorems of this article are contributions to invertibility in the context of\u0000ordered and weak-quasi-topological rings. Specifically, the first theorem\u0000asserts that the interval $]0,1]$ in any suitable partially ordered ring\u0000consists entirely of invertible elements. The second theorem asserts that if\u0000$f$ is a norm from a ring to a partially ordered ring endowed with interval\u0000topology, then under certain conditions, the subset of elements such that\u0000$f(1-a) < 1$ consists entirely of invertible elements. The second theorem\u0000relies on the assumption of sequential Cauchy completeness of the topology\u0000induced by the norm $f$, which as we recall, takes values in an ordered ring\u0000endowed with the interval topology (an example of a coarse topology). The fact\u0000that a ring endowed with the topology associated with a seminorm into an\u0000ordered ring endowed with the interval topology is a locally convex\u0000quasi-topological group with an additional continuity property of the product\u0000is dealt with in a separate section. A brief application to frame theory is\u0000also included.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"129 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141168522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","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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