In this manuscript, a recent topology on the positive integers, generated by the collection of relatively prime positive integers, 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), different from those presented in the style of H. Furstenberg. Finally, some problems are proposed.
在本手稿中，由相对素正整数集合生成的正整数上的最新拓扑学被推广到积分域上。研究了它的一些拓扑性质。此外，还探讨了这种拓扑学在非域的无限主理想域上的性质，并获得了素元无穷大的新拓扑学证明（假设单位集是有限的），这与 H. Furstenberg 风格的证明不同。最后，还提出了一些问题。
{"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}
Mohamed Jleli, Cristina Maria Pacurar, Bessem Samet
We introduce two new classes of single-valued contractions of polynomial type defined on a metric space. For the first one, called the class of polynomial contractions, we establish two fixed point theorems. Namely, we first consider the case when the mapping is continuous. Next, we weaken the continuity condition. In particular, we recover Banach's fixed point theorem. The second class, called the class of almost polynomial contractions, includes the class of almost contractions introduced by Berinde [Nonlinear Analysis Forum. 9(1) (2004) 43--53]. A fixed point theorem is established for almost polynomial contractions. The obtained result generalizes that derived by Berinde in the above reference. Several examples showing that our generalizations are significant, are provided.
{"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}
A space $X$ is sequentially separable if there is a countable $Ssubset X$ 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 sequentially separable: $sigma$-separability and $F$-separability. The aim of this paper is to study $sigma$-separability and $F$-separability (and their hereditary variants) of the space $C_p(X)$ of all real-valued continuous functions, defined on a Tychonoff space $X$, endowed with the pointwise convergence topology. In particular, we proved that $sigma$-separability coincides with sequential separability. Hereditary variants (hereditarily $sigma$-separablity and hereditarily $F$-separablity) coincides with Frechet-Urysohn property in the class of cosmic spaces.
{"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}
We develop a minimal system to study the stochastic formation of Borromean links within topologically entangled networks without requiring the use of knot invariants. Borromean linkages may form in entangled solutions of open polymer chains or in Olympic gel systems such as kinetoplast DNA, but it is challenging to investigate this due to the difficulty of computing three-body link invariants. Here, we investigate randomly oriented rectangles densely packed within a volume, and evaluate them for Hopf linking and Borromean link formation. We show that dense packings of rectangles can form Borromean triplets and larger clusters, and that in high enough density the combination of Hopf and Borromean linking can create a percolating hypergraph through the network. We present data for the percolation threshold of Borromean hypergraphs, and discuss implications for the existence of Borromean connectivity within kinetoplast DNA.
我们开发了一个最小系统来研究在拓扑纠缠网络中随机形成的波罗曼（Borromean）链接，而不需要使用节点变量。在开放聚合物链的纠缠溶液或奥林匹克凝胶系统（如动原 DNA）中可能会形成博罗梅因链接，但由于难以计算三体链接变量，研究这种情况具有挑战性。在这里，我们研究了在一个体积内密集堆积的随机取向矩形，并对它们进行了霍普夫链接和博罗曼链接变换评估。我们的研究表明，密集堆积的矩形可以形成 Borromeantriplets 和更大的簇，而且在足够高的密度下，Hopf 链接和 Borromean 链接的组合可以在网络中形成一个渗滤超图。我们提出了博罗梅斯超图的渗流阈值数据，并讨论了动粒 DNA 中存在博罗梅斯连通性的意义。
{"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}
In this document, we propose a bridge between the graphs and the geometric realizations of their Vietoris Rips complexes, i.e. Graphs, with their canonical v{C}ech closure structure, have the same homotopy type that the realization of their Vietoris Rips complex.
{"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}
In this paper we study interval maps with zero topological entropy that are crooked; i.e. whose inverse limit is the pseudo-arc. We show that there are uncountably many pairwise non-conjugate zero entropy crooked interval maps with different sets of fixed points. We also show that there are uncountably many crooked maps that are pairwise non-conjugate and have exactly two fixed points. Furthermore, we provide a characterization of crooked interval maps that are under (above) the diagonal.
{"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}
We show that lattice isomorphisms between the lattices of slowly oscillating functions on chain-connected proper metric spaces induce coarsely equivalent homeomorphisms. This result leads to a Banach-Stone-type theorem for these lattices. Furthermore, we provide a representation theorem that characterizes linear lattice isomorphisms among these lattices.
{"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}
G. Bezhanishvili, F. Dashiell Jr, A. Moshier, J. Walters-Wayland
Completions play an important r^ole for studying structure by supplying elements that in some sense ``ought to be." Among these, the Dedekind-MacNeille completion is of particular importance. In 1968 Janowitz provided necessary and sufficient conditions for it to be subfit or Boolean. Another natural separation axiom situated between the two is regularity. We explore similar characterizations of when closely related completions are subfit, regular, or Boolean. We are mainly interested in the Bruns-Lakser, ideal, and canonical completions, which are useful in pointfree topology since (unlike the Dedekind-MacNeille completion) they satisfy stronger forms of distributivity.
{"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}
A finite family $mathcal{F}={f_1,ldots,f_n}$ of continuous selfmaps of a given metric space $X$ is called an iterated function system (shortly IFS). In a case of contractive selfmaps of a complete metric space is well-known that IFS has an unique attractor cite{Hu}. However, in cite{LS} authors studied highly non-contractive IFSs, i.e. such families $mathcal{F}={f_1,ldots,f_n}$ of continuous selfmaps that for any remetrization of $X$ each function $f_i$ has Lipschitz constant $>1, i=1,ldots,n.$ They asked when one can remetrize $X$ that $mathcal{F}$ is Lipschitz IFS, i.e. all $f_i's$ are Lipschitz (not necessarily contractive), $ i=1,ldots,n$. We give a general positive answer for this problem by constructing respective new metric (equivalent to the original one) on $X$, determined by a given family $mathcal{F}={f_1,ldots,f_n}$ of continuous selfmaps of $X$. However, our construction is valid even for some specific infinite families of continuous functions.
{"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}
Invertibility is important in ring theory because it enables division and facilitates solving equations. Moreover, rings can be endowed with extra ''structure'' such as order and topology that add new properties. The two main theorems of this article are contributions to invertibility in the context of ordered and weak-quasi-topological rings. Specifically, the first theorem asserts that the interval $]0,1]$ in any suitable partially ordered ring consists entirely of invertible elements. The second theorem asserts that if $f$ is a norm from a ring to a partially ordered ring endowed with interval topology, then under certain conditions, the subset of elements such that $f(1-a) < 1$ consists entirely of invertible elements. The second theorem relies on the assumption of sequential Cauchy completeness of the topology induced by the norm $f$, which as we recall, takes values in an ordered ring endowed with the interval topology (an example of a coarse topology). The fact that a ring endowed with the topology associated with a seminorm into an ordered ring endowed with the interval topology is a locally convex quasi-topological group with an additional continuity property of the product is dealt with in a separate section. A brief application to frame theory is also included.
{"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}
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