{"title":"Zero-Set Intersection Graph On C+(X)","authors":"Soumi Basu, Bedanta Bose","doi":"arxiv-2407.08235","DOIUrl":"https://doi.org/arxiv-2407.08235","url":null,"abstract":"For any Tychonoff space X we have introduced the zero-set in-tersection graph\u0000on {Gamma}(C+(X)) and studied the graph properties in connection with the\u0000algebraic properties of the semiring C+(X). We have shown that for any two\u0000realcompact spaces X and Y the graph isomorphism between {Gamma}(C+(X)) and\u0000{Gamma}(C+(Y )), the semiring isomorphism between C+(X) and C+(Y ), the\u0000topological homeomorphism between X and Y, the ring isomorphism between C(X)\u0000and C(Y ) and the graph isomorphism between {Gamma}(C(X)) and {Gamma}(C(Y ))\u0000are equivalent.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141611951","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}
{"title":"On Stronger Forms of Expansivity","authors":"Shital H. Joshi, Ekta Shah","doi":"arxiv-2407.07549","DOIUrl":"https://doi.org/arxiv-2407.07549","url":null,"abstract":"We define the concept of stronger forms of positively expansive map and name\u0000it as $p :mathscr{F}-$expansive maps. Here $mathscr{F}$ is a family of\u0000subsets of $mathbb{N}$. Examples of positively thick expansive and positively\u0000syndetic expansive maps are constructed here. Also, we obtain conditions under\u0000which a positively expansive map is positively co--finite expansive and\u0000positively syndetic expansive maps. Further, we study several properties of $p\u0000:mathscr{F}-$expansive maps. A characterization of $p\u0000:mathscr{F}-$expansive maps in terms of $p :mathscr{F}^*-$generator is\u0000obtained. Here $p :mathscr{F}^*$ is dual of $mathscr{F}$. Considering\u0000$(mathbb{Z},+)$ as a semigroup, we study $mathscr{F}-$expansive\u0000homeomorphism, where $mathscr{F}$ is a family of subsets of $mathbb{Z}\u0000setminus {0}$. We show that there does not exists an expansive homeomorphism\u0000on a compact metric space which is $mathscr{F}_s-$expansive. Also, we study\u0000relation between $mathscr{F}-$expansivity of $f$ and the shift map $sigma_f$\u0000on the inverse limit space.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141588598","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}
{"title":"Statistical convergence in metric-like spaces","authors":"Prasanta Malik, Saikat Das","doi":"arxiv-2407.07117","DOIUrl":"https://doi.org/arxiv-2407.07117","url":null,"abstract":"In this paper we introduce the notions of statistical convergence and\u0000statistical Cauchyness of sequences in a metric-like space. We study some basic\u0000properties of these notions","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141587101","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}
{"title":"Virtual Lie subgroups of locally compact groups","authors":"Antoni Machowski","doi":"arxiv-2407.01253","DOIUrl":"https://doi.org/arxiv-2407.01253","url":null,"abstract":"We examine subgroups of locally compact groups that are continuous\u0000homomorphic images of connected Lie groups and we give a criterion for being\u0000such an image. We also provide a new characterisation of Lie groups and a\u0000characterisation of groups that are images of connected locally compact groups.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501273","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}
{"title":"Delhomme-Laflamme-Pouzet-Sauer space as groupoid","authors":"Oleksiy Dovgoshey, Alexander Kostikov","doi":"arxiv-2407.00508","DOIUrl":"https://doi.org/arxiv-2407.00508","url":null,"abstract":"Let $mathbb{R}^{+}=[0, infty)$ and let $d^+$ be the ultrametric on\u0000$mathbb{R}^+$ such that $d^+ (x,y) = max{x,y}$ for all different $x,y in\u0000mathbb{R}^+$. It is shown that the monomorphisms of the groupoid\u0000$(mathbb{R}^+, d^+)$ coincide with the injective ultrametric-preserving\u0000functions and that the automorphisms of $(mathbb{R}^+, d^+)$ coincide with the\u0000self-homeomorphisms of $mathbb{R}^+$. The structure of endomorphisms of\u0000$(mathbb{R}^+, d^+)$ is also described.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141524743","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}
{"title":"Separation of horocycle orbits on moduli space in genus 2","authors":"John Rached","doi":"arxiv-2406.19527","DOIUrl":"https://doi.org/arxiv-2406.19527","url":null,"abstract":"We prove a quantitative closing lemma for the horocycle flow induced by the\u0000$mathrm{SL}(2,mathbb{R})$-action on the moduli space of Abelian differentials\u0000with a double-order zero on surfaces of genus 2. The proof proceeds via\u0000construction of a Margulis function measuring the discretized fractal dimension\u0000of separation of a horocycle orbit of a point from itself, in a direction\u0000transverse to the $mathrm{SL}(2,mathbb{R})$-orbit. From this, we deduce that\u0000small transversal separation guarantees the existence of a nearby point with a\u0000pseudo-Anosov in its Veech group. This is reminiscent of the initial dimension\u0000phases in Bourgain-Gamburd for random walks on compact groups,\u0000Bourgain-Lindenstrauss-Furman-Mozes for quantitative equidistribution in tori,\u0000and quantitative equidistribution of horocycle flow for a product of\u0000$mathrm{SL}(2,mathbb{R})$ with itself due to Lindenstrauss-Mohammadi-Wang,\u0000and multiple other works.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141524745","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}
{"title":"High dimensional countable compactness and ultrafilters","authors":"Cesar Corral, Pourya Memarpanahi, Paul Szeptycki","doi":"arxiv-2406.17217","DOIUrl":"https://doi.org/arxiv-2406.17217","url":null,"abstract":"We define several notions of a limit point on sequences with domain a barrier\u0000in $[omega]^{<omega}$ focusing on the two dimensional case $[omega]^2$. By\u0000exploring some natural candidates, we show that countable compactness has a\u0000number of generalizations in terms of limits of high dimensional sequences and\u0000define a particular notion of $alpha$-countable compactness for\u0000$alphaleqomega_1$. We then focus on dimension 2 and compare 2-countable\u0000compactness with notions previously studied in the literature. We present a\u0000number of counterexamples showing that these classes are different. In\u0000particular assuming the existence of a Ramsey ultrafilter, a subspace of\u0000$betaomega$ which is doubly countably compact whose square is not countably\u0000compact, answering a question of T. Banakh, S. Dimitrova and O. Gutik. The\u0000analysis of this construction leads to some possibly new types of ultrafilters\u0000related to discrete, P-points and Ramsey ultrafilters.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501275","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}
Guojun WuSchool of Mathematics and Statistics, Nanjing University of Information Science and TechnologyApplied Mathematics Center of Jiangsu Province, Nanjing University of Information Science and Technology, Wei YaoSchool of Mathematics and Statistics, Nanjing University of Information Science and TechnologyApplied Mathematics Center of Jiangsu Province, Nanjing University of Information Science and Technology, Qingguo LiSchool of Mathematics, Hunan University
{"title":"Representations of domains via closure spaces in the quantale-valued setting","authors":"Guojun WuSchool of Mathematics and Statistics, Nanjing University of Information Science and TechnologyApplied Mathematics Center of Jiangsu Province, Nanjing University of Information Science and Technology, Wei YaoSchool of Mathematics and Statistics, Nanjing University of Information Science and TechnologyApplied Mathematics Center of Jiangsu Province, Nanjing University of Information Science and Technology, Qingguo LiSchool of Mathematics, Hunan University","doi":"arxiv-2406.17712","DOIUrl":"https://doi.org/arxiv-2406.17712","url":null,"abstract":"With a commutative unital quantale $L$ as the truth value table, this study\u0000focuses on the representations of $L$-domains by means of $L$-closure spaces.\u0000First, the notions of interpolative generalized $L$-closure spaces and directed\u0000closed sets are introduced. It is proved that in an interpolative generalized\u0000$L$-closure space (resp., $L$-closure space), the collection of directed closed\u0000sets with respect to the inclusion $L$-order forms a continuous $L$-dcpo\u0000(resp., an algebraic $L$-dcpo). Conversely, it is shown that every continuous\u0000$L$-dcpo (resp., algebraic $L$-dcpo) can be reconstructed by an interpolative\u0000generalized $L$-closure space (resp., $L$-closure space). Second, when $L$ is\u0000integral, the notion of dense subspaces of generalized $L$-closure spaces is\u0000introduced. By means of dense subspaces, an alternative representation for\u0000algebraic $L$-dcpos is given. Moreover, the concept of $L$-approximable\u0000relations between interpolative generalized $L$-closure spaces is introduced.\u0000Consequently, a categorical equivalence between the category of interpolative\u0000generalized $L$-closure spaces (resp., $L$-closure spaces) with\u0000$L$-approximable relations and that of continuous $L$-dcpos (resp., algebraic\u0000$L$-dcpos) with Scott continuous mappings is established.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501274","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}
{"title":"Center and radius of a subset of metric space","authors":"Akhilesh Badra, Hemant Kumar Singh","doi":"arxiv-2406.15772","DOIUrl":"https://doi.org/arxiv-2406.15772","url":null,"abstract":"In this paper, we introduce a notion of the center and radius of a subset A\u0000of metric space X. In the Euclidean spaces, this notion can be seen as the\u0000extension of the center and radius of open/closed balls. The center and radius\u0000of a finite product of subsets of metric spaces, and a finite union of subsets\u0000of a metric space are also determined. For any subset A of metric space X,\u0000there is a natural question to identify the open balls of X with the largest\u0000radius that are entirely contained in A. To answer this question, we introduce\u0000a notion of quasi-center and quasi-radius of a subset A of metric space X. We\u0000prove that the center of the largest open balls contained in A belongs to the\u0000quasi-center of A, and its radius is equal to the quasi-radius of A. In\u0000particular, for the Euclidean spaces, we see that the center of largest open\u0000balls contained in A belongs to the center of A, and its radius is equal to the\u0000radius of A.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501276","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}
{"title":"Chessboard and level sets of continuous functions","authors":"Michał Dybowski, Przemysław Górka","doi":"arxiv-2406.13774","DOIUrl":"https://doi.org/arxiv-2406.13774","url":null,"abstract":"We show the following result: Let $f colon I^n to mathbb{R}^{n-1}$ be a\u0000continuous function. Then, there exist $p in mathbb{R}^{n-1}$ and compact\u0000subset $S subset f^{-1}left[left{pright}right]$ which connects some\u0000opposite faces of the $n$-dimensional unit cube $I^n$. We give an example that\u0000shows it cannot be generalized to path-connected sets. We also provide a\u0000discrete version of this result which is inspired by the $n$-dimensional\u0000Steinhaus Chessboard Theorem. Additionally, we show that the latter one and the\u0000Brouwer Fixed Point Theorem are simple consequences of the main result.","PeriodicalId":501314,"journal":{"name":"arXiv - MATH - General Topology","volume":"78 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501278","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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