On the Topology $\tau_{R}^{\diamond}$ of Primal Topological Spaces

On the Topology $\tau_{R}^{\diamond}$ of Primal Topological Spaces | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article On the Topology $\tau_{R}^{\diamond}$ of Primal Topological Spaces Murad ÖZKOÇ, Büşra KÖSTEL This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3962079/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The main purpose of this paper is to introduce and study two new operators $(\cdot)_{R}^{\diamond}$ and $cl_{R}^{\diamond}(\cdot)$ via primal which is a new notion. We also show that the operator $cl_{R}^{\diamond}(\cdot)$ is a Kuratowski closure operator, while the operator $(\cdot)_{R}^{\diamond}$ is not. In addition, we prove that the topology on $X$, shown as $\tau_{R}^{\diamond}$, obtained by means of the operator $cl_{R}^{\diamond}(\cdot)$ is finer than $\tau_{\delta}$, where $\tau_{\delta}$ is the family of $\delta$-open subsets of a space $(X,\tau)$. Moreover, we not only obtain a base for the topology $\tau_{R}^{\diamond}$ but also prove many fundamental results concerning this new structure. Furthermore, we give many counterexamples related to our results. 2020 AMS Classifications: 54A05; 54B99; 54C60. Topology Primal primal topological space Kuratowski closure operator the operator $(\cdot)_{R}^{\diamond}$ the operator $cl_{R}^{\diamond}$ Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3962079","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":273269414,"identity":"496c61bf-873c-491c-b634-165eb4a49dc3","order_by":0,"name":"Murad ÖZKOÇ","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA2UlEQVRIiWNgGAWjYLACHiCWYGBgfABi85GihdkAxGYjRQubBIhDUIvujNyHH94w2NhLtjc/q/yaYyfDxsD88NENPFrMbqQbS85hSGOW5jlmdlt2WzLQYWzGxjl4taQxSPMwHGaTk8hhuy25jRmohYdNmoAW5t9ALTwgLcWS2+qJ0sIGskVCGqiF8eO2w0RoOfOMzXKOQZqBZM8xY2nGbcd52JgJ+eV4GvONNxU29hLHmx9+/Lmt2p6fvfnhY3xaIMAAQjHzgEmCypEA4w9SVI+CUTAKRsGIAQCYNztn3ibKlwAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0003-0068-7415","institution":"Muğla Sıtkı Koçman University","correspondingAuthor":true,"prefix":"","firstName":"Murad","middleName":"","lastName":"ÖZKOÇ","suffix":""},{"id":273269520,"identity":"6bb633fb-cb9c-48ff-b174-66968747b3f6","order_by":1,"name":"Büşra KÖSTEL","email":"","orcid":"","institution":"Muğla Sıtkı Koçman University","correspondingAuthor":false,"prefix":"","firstName":"Büşra","middleName":"","lastName":"KÖSTEL","suffix":""}],"badges":[],"createdAt":"2024-02-16 18:45:24","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false,"coiExplicitlySet":false},"doi":"10.21203/rs.3.rs-3962079/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3962079/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":51322129,"identity":"e42928bb-b4a4-425f-bd4a-7f34df280e99","added_by":"auto","created_at":"2024-02-19 14:47:42","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":241179,"visible":true,"origin":"","legend":"","description":"","filename":"SOCOD240089016.02.202411.45.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3962079/v1_covered_6d08be0b-b6af-43bd-a791-657b72892ce0.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eOn the Topology $\\tau_{R}^{\\diamond}$ of Primal Topological Spaces\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Muğla Sıtkı Koçman University","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Primal, primal topological space, Kuratowski closure operator, the operator $(\\cdot)_{R}^{\\diamond}$, the operator $cl_{R}^{\\diamond}$","lastPublishedDoi":"10.21203/rs.3.rs-3962079/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3962079/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe main purpose of this paper is to introduce and study two new operators $(\\cdot)_{R}^{\\diamond}$\u0026nbsp;and $cl_{R}^{\\diamond}(\\cdot)$\u0026nbsp;via primal which is a new notion. We also show that the operator $cl_{R}^{\\diamond}(\\cdot)$ is a Kuratowski closure operator, while the operator $(\\cdot)_{R}^{\\diamond}$ is not. In addition, we prove that the topology on $X$, shown as $\\tau_{R}^{\\diamond}$, obtained by means of the operator $cl_{R}^{\\diamond}(\\cdot)$ is finer than $\\tau_{\\delta}$, where $\\tau_{\\delta}$\u0026nbsp;is the family of $\\delta$-open subsets of a space $(X,\\tau)$. Moreover, we not only obtain a base for the topology $\\tau_{R}^{\\diamond}$ but also prove many fundamental results concerning this new structure.\u0026nbsp;Furthermore, we give many counterexamples related to our results.\u003c/p\u003e\n\u003cp\u003e2020 AMS Classifications: 54A05; 54B99; 54C60.\u003c/p\u003e","manuscriptTitle":"On the Topology $\\tau_{R}^{\\diamond}$ of Primal Topological Spaces","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-02-19 14:39:33","doi":"10.21203/rs.3.rs-3962079/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"c509a040-5abb-485c-a25e-a33971d30fc0","owner":[],"postedDate":"February 19th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":28809772,"name":"Topology"}],"tags":[],"updatedAt":"2024-02-19T14:39:33+00:00","versionOfRecord":[],"versionCreatedAt":"2024-02-19 14:39:33","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3962079","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3962079","identity":"rs-3962079","version":["v1"]},"buildId":"_2-kVJe1T_tPrBINL-cwx","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}