We define a emph{residual function} on a topological space $X$ as a function $f:Xlongrightarrowmathbb{Z}$ such that $f^{-1}(0)$ contains an open dense set, and we use this notion to study the freeness of the group of divisorial ideals on a Pr"ufer domain.
{"title":"Residual functions and divisorial ideals","authors":"Dario Spirito","doi":"arxiv-2409.11846","DOIUrl":"https://doi.org/arxiv-2409.11846","url":null,"abstract":"We define a emph{residual function} on a topological space $X$ as a function\u0000$f:Xlongrightarrowmathbb{Z}$ such that $f^{-1}(0)$ contains an open dense\u0000set, and we use this notion to study the freeness of the group of divisorial\u0000ideals on a Pr\"ufer domain.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258562","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 associate a new topology to a nonzero unital module $M$ over a commutative $R$, which is called Golomb topology of the $R$-module $M$. Let $M $be an $R$-module and $B_{M}$ be the family of coprime cosets ${m+N}$ where $min M$ and $N $is a nonzero submodule of $M $such that $N+Rm=M$. We prove that if $M $is a meet irreducible multiplication module or $M $is a meet irreducible finitely generated module in which every maximal submodule is strongly irreducible, then $B_{M} $is the basis for a topology on $M $which is denoted by $widetilde{G(M)}.$ In particular, the subspace topology on $M-{0}$ is called the Golomb topology of the $R$-module $M $and denoted by $G(M)$. We investigate the relations between topological properties of $G(M) $and algebraic properties of $M. $In particular, we characterize some important classes of modules such as simple modules, Jacobson semisimple modules in terms of Golomb topology.
{"title":"On Golomb Topology of Modules over Commutative Rings","authors":"Uğur Yiğit, Suat Koç, Ünsal Tekir","doi":"arxiv-2409.09807","DOIUrl":"https://doi.org/arxiv-2409.09807","url":null,"abstract":"In this paper, we associate a new topology to a nonzero unital module $M$\u0000over a commutative $R$, which is called Golomb topology of the $R$-module $M$.\u0000Let $M $be an $R$-module and $B_{M}$ be the family of coprime cosets\u0000${m+N}$ where $min M$ and $N $is a nonzero submodule of $M $such that\u0000$N+Rm=M$. We prove that if $M $is a meet irreducible multiplication module or\u0000$M $is a meet irreducible finitely generated module in which every maximal\u0000submodule is strongly irreducible, then $B_{M} $is the basis for a topology on\u0000$M $which is denoted by $widetilde{G(M)}.$ In particular, the subspace\u0000topology on $M-{0}$ is called the Golomb topology of the $R$-module $M $and\u0000denoted by $G(M)$. We investigate the relations between topological properties\u0000of $G(M) $and algebraic properties of $M. $In particular, we characterize\u0000some important classes of modules such as simple modules, Jacobson semisimple\u0000modules in terms of Golomb topology.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258565","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}
Let $R $be an integral domain and $R^{#}$ the set of all nonzero nonunits of $R. $For every elements $a,bin R^{#},$ we define $asim b$ if and only if $aR=bR,$ that is, $a$ and $b$ are associated elements. Suppose that $EC(R^{#})$ is the set of all equivalence classes of $R^{#} $according to $sim$.$ $Let $U_{a}={[b]in EC(R^{#}):b $divides $a}$ for every $ain R^{#}.$ Then we prove that the family ${U_{a}}_{ain R^{#}}$ becomes a basis for a topology on $EC(R^{#}). $This topology is called divisor topology of $R $and denoted by $D(R). $We investigate the connections between the algebraic properties of $R $and the topological properties of$ D(R)$. In particular, we investigate the seperation axioms on $D(R)$, first and second countability axioms, connectivity and compactness on $D(R)$. We prove that for atomic domains $R, $the divisor topology $D(R) $is a Baire space. Also, we characterize valution domains $R$ in terms of nested property of $D(R).$ In the last section, we introduce a new topological proof of the infinitude of prime elements in a UFD and integers by using the topology $D(R)$.
对于R^{/#}中的每个元素$a,b,$我们定义$a/sim b$，当且仅当$aR=bR,$即$a$和$b$是关联元素。假设$EC(R^{/#})$是$R^{/#}/$的所有等价类的集合，根据$a/sim$.$让$U_{a}=/{[b]in EC(R^{#}):b$divides $a$ for every $ainR^{#}.然后我们证明${U_{a}}_{ain R^{#}}$ 系列在$EC(R^{/#})上的拓扑学中变得无足轻重。 $We research the connections between thealgebraic properties of $R$ and the topological properties of $D(R)$.特别是，我们研究了$D(R)$上的分离公理、第一可数公理和第二可数公理、连通性和紧凑性。我们证明，对于原子域 $R,$，除数拓扑 $D(R)$ 是一个拜尔空间。在最后一节中，我们利用拓扑 $D(R)$ 介绍了 UFD 和整数中原元无穷大的新拓扑证明。
{"title":"On Divisor Topology of Commutative Rings","authors":"Uğur Yiğit, Suat Koç","doi":"arxiv-2409.10577","DOIUrl":"https://doi.org/arxiv-2409.10577","url":null,"abstract":"Let $R $be an integral domain and $R^{#}$ the set of all nonzero nonunits\u0000of $R. $For every elements $a,bin R^{#},$ we define $asim b$ if and only if\u0000$aR=bR,$ that is, $a$ and $b$ are associated elements. Suppose that\u0000$EC(R^{#})$ is the set of all equivalence classes of $R^{#} $according to\u0000$sim$.$ $Let $U_{a}={[b]in EC(R^{#}):b $divides $a}$ for every $ain\u0000R^{#}.$ Then we prove that the family ${U_{a}}_{ain R^{#}}$ becomes a\u0000basis for a topology on $EC(R^{#}). $This topology is called divisor topology\u0000of $R $and denoted by $D(R). $We investigate the connections between the\u0000algebraic properties of $R $and the topological properties of$ D(R)$. In\u0000particular, we investigate the seperation axioms on $D(R)$, first and second\u0000countability axioms, connectivity and compactness on $D(R)$. We prove that for\u0000atomic domains $R, $the divisor topology $D(R) $is a Baire space. Also, we\u0000characterize valution domains $R$ in terms of nested property of $D(R).$ In the\u0000last section, we introduce a new topological proof of the infinitude of prime\u0000elements in a UFD and integers by using the topology $D(R)$.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258563","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}
The paper contains a very simple proof of the classical Hasumi's theorem that each usco mapping defined on an extremally disconnected space has a continuous selection. The paper also contains a very simple proof of a recent result about extension of densely defined continuous selections for compact-valued continuous mappings, in fact a generalisation of this result to all usco mappings with a regular range.
{"title":"Two Selection Theorems for Extremally Disconnected Spaces","authors":"Valentin Gutev","doi":"arxiv-2409.09490","DOIUrl":"https://doi.org/arxiv-2409.09490","url":null,"abstract":"The paper contains a very simple proof of the classical Hasumi's theorem that\u0000each usco mapping defined on an extremally disconnected space has a continuous\u0000selection. The paper also contains a very simple proof of a recent result about\u0000extension of densely defined continuous selections for compact-valued\u0000continuous mappings, in fact a generalisation of this result to all usco\u0000mappings with a regular range.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258564","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}
The initial part of this paper is devoted to the notion of pseudo-seminorm on a vector space $E$. We prove that the topology of every topological vector space is defined by a family of pseudo-seminorms (and so, as it is known, it is uniformizable). Then we devote ourselves to the Lipschitz vector structures on $E$, that is those Lipschitz structures on $E$ for which the addition is a Lipschitz map, while the scalar multiplication is a locally Lipschitz map, and we prove that any topological vector structure on $E$ is associated to some Lipschitz vector structure. Afterwards, we attend to the bornological Lipschitz maps. The final part of the article is devoted to the Lipschitz vector structures compatible with locally convex topologies on $E$.
{"title":"Lipschitz vector spaces","authors":"Tullio Valent","doi":"arxiv-2409.06574","DOIUrl":"https://doi.org/arxiv-2409.06574","url":null,"abstract":"The initial part of this paper is devoted to the notion of pseudo-seminorm on\u0000a vector space $E$. We prove that the topology of every topological vector\u0000space is defined by a family of pseudo-seminorms (and so, as it is known, it is\u0000uniformizable). Then we devote ourselves to the Lipschitz vector structures on\u0000$E$, that is those Lipschitz structures on $E$ for which the addition is a\u0000Lipschitz map, while the scalar multiplication is a locally Lipschitz map, and\u0000we prove that any topological vector structure on $E$ is associated to some\u0000Lipschitz vector structure. Afterwards, we attend to the bornological Lipschitz maps. The final part of\u0000the article is devoted to the Lipschitz vector structures compatible with\u0000locally convex topologies on $E$.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181045","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 topological space $X$ is Baire if the Baire Category Theorem holds for $X$, i.e., the intersection of any sequence of open dense subsets of $X$ is dense in $X$. One of the interesting problems in the theory of functional spaces is the characterization of the Baire property of a functional space through the topological property of the support of functions. In the paper this problem is solved for the space $M(X, K)$ of all measurable compact-valued ($K$-valued) functions defined on a measurable space $(X,Sigma)$ with the topology of pointwise convergence. It is proved that $M(X, K)$ is Baire for any metrizable compact space $K$.
{"title":"On Baire property of spaces of compact-valued measurable functions","authors":"Alexander V. Osipov","doi":"arxiv-2409.02913","DOIUrl":"https://doi.org/arxiv-2409.02913","url":null,"abstract":"A topological space $X$ is Baire if the Baire Category Theorem holds for $X$,\u0000i.e., the intersection of any sequence of open dense subsets of $X$ is dense in\u0000$X$. One of the interesting problems in the theory of functional spaces is the\u0000characterization of the Baire property of a functional space through the\u0000topological property of the support of functions. In the paper this problem is solved for the space $M(X, K)$ of all measurable\u0000compact-valued ($K$-valued) functions defined on a measurable space\u0000$(X,Sigma)$ with the topology of pointwise convergence. It is proved that\u0000$M(X, K)$ is Baire for any metrizable compact space $K$.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181048","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 prove that a $T_0$ topological space is $omega$-well-filtered if and only if it does not admit either the natural numbers with the cofinite topology or with the Scott topology as its closed subsets in the strong topology. Based on this, we offer a refined topological characterization for the $omega$-well-filterification of $T_0$-spaces and solve a problem posed by Xiaoquan Xu. In the setting of second countable spaces, we also characterise sobriety by convergences of certain $Pi^0_2$-Cauchy subsets of the spaces.
{"title":"$ω$-well-filtered spaces, revisited","authors":"Hualin Miao, Xiaodong Jia, Ao Shen, Qingguo Li","doi":"arxiv-2409.01551","DOIUrl":"https://doi.org/arxiv-2409.01551","url":null,"abstract":"We prove that a $T_0$ topological space is $omega$-well-filtered if and only\u0000if it does not admit either the natural numbers with the cofinite topology or\u0000with the Scott topology as its closed subsets in the strong topology. Based on\u0000this, we offer a refined topological characterization for the\u0000$omega$-well-filterification of $T_0$-spaces and solve a problem posed by\u0000Xiaoquan Xu. In the setting of second countable spaces, we also characterise\u0000sobriety by convergences of certain $Pi^0_2$-Cauchy subsets of the spaces.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181046","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 prove analogues of classical results for higher homotopy groups and singular homology groups of pseudotopological spaces. Pseudotopological spaces are a generalization of (v{C}ech) closure spaces which are in turn a generalization of topological spaces. Pseudotopological spaces also include graphs and directed graphs as full subcategories. Thus they are a bridge that connects classical algebraic topology with the more applied side of topology. More specifically, we show the existence of a long exact sequence for homotopy groups of pairs of pseudotopological spaces and that a weak homotopy equivalence induces isomorphisms for homology groups. Our main result is the construction of weak homotopy equivalences between the geometric realizations of directed Vietoris-Rips complexes and their underlying directed graphs. This implies that singular homology groups of finite directed graphs can be efficiently calculated from finite combinatorial structures, despite their associated chain groups being infinite dimensional. This work is similar to the work of McCord for finite topological spaces but in the context of pseudotopological spaces. Our results also give a novel approach for studying (higher) homotopy groups of discrete mathematical structures such as (directed) graphs or digital images.
{"title":"The directed Vietoris-Rips complex and homotopy and singular homology groups of finite digraphs","authors":"Nikola Milićević, Nicholas A. Scoville","doi":"arxiv-2409.01370","DOIUrl":"https://doi.org/arxiv-2409.01370","url":null,"abstract":"We prove analogues of classical results for higher homotopy groups and\u0000singular homology groups of pseudotopological spaces. Pseudotopological spaces\u0000are a generalization of (v{C}ech) closure spaces which are in turn a\u0000generalization of topological spaces. Pseudotopological spaces also include\u0000graphs and directed graphs as full subcategories. Thus they are a bridge that\u0000connects classical algebraic topology with the more applied side of topology.\u0000More specifically, we show the existence of a long exact sequence for homotopy\u0000groups of pairs of pseudotopological spaces and that a weak homotopy\u0000equivalence induces isomorphisms for homology groups. Our main result is the\u0000construction of weak homotopy equivalences between the geometric realizations\u0000of directed Vietoris-Rips complexes and their underlying directed graphs. This\u0000implies that singular homology groups of finite directed graphs can be\u0000efficiently calculated from finite combinatorial structures, despite their\u0000associated chain groups being infinite dimensional. This work is similar to the\u0000work of McCord for finite topological spaces but in the context of\u0000pseudotopological spaces. Our results also give a novel approach for studying\u0000(higher) homotopy groups of discrete mathematical structures such as (directed)\u0000graphs or digital images.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181047","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 deal with non-Baire rare sets in category bases which forms $aleph_0$-independent family, where a rare set is a common generalization of both Luzin and Sierpinski set.
{"title":"On non-Baire rare sets in category bases","authors":"Sanjib Basu, Abhit Chandra Pramanik","doi":"arxiv-2409.01430","DOIUrl":"https://doi.org/arxiv-2409.01430","url":null,"abstract":"In this paper, we deal with non-Baire rare sets in category bases which forms\u0000$aleph_0$-independent family, where a rare set is a common generalization of\u0000both Luzin and Sierpinski set.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181049","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, using the concept of natural density, we have introduced the ideas of statistical and rough statistical convergence in an $S$-metric space. We have investigated some of their basic properties. We have defined statistical Cauchyness and statistical boundedness of sequences and then some results related these ideas have been studied. We have defined the set of rough statistical limit points of a sequence in an $S$-metric space and have proved some relevant results associated with such type of convergence
{"title":"Statistical and rough statistical convergence in an S-metric space","authors":"Sukila Khatun, Amar Kumar Banerjee","doi":"arxiv-2408.14973","DOIUrl":"https://doi.org/arxiv-2408.14973","url":null,"abstract":"In this paper, using the concept of natural density, we have introduced the\u0000ideas of statistical and rough statistical convergence in an $S$-metric space.\u0000We have investigated some of their basic properties. We have defined\u0000statistical Cauchyness and statistical boundedness of sequences and then some\u0000results related these ideas have been studied. We have defined the set of rough\u0000statistical limit points of a sequence in an $S$-metric space and have proved\u0000some relevant results associated with such type of convergence","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181051","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：