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window.loswp.shouldDetectTimezone = true; window.loswp.shouldShowBulkDownload = true; window.loswp.showSignupCaptcha = false window.loswp.willEdgeCache = false; window.loswp.work = {"work":{"id":14538401,"created_at":"2015-07-31T13:57:09.283-07:00","from_world_paper_id":null,"updated_at":"2024-11-30T05:38:31.261-08:00","_data":{"ai_title_tag":"αγ-Open Sets and Continuous Mappings in Topological Spaces","grobid_abstract":"In this paper, we introduce the concept of α γ -open sets as a generalization of γ-open sets in a topological space (X, τ). Using this set, we introduce α γ T 0 , α γ -T ½ , α γ T 1 , α γ T 2 , α γ D 0 , α γ D 1 and α γ D 2 spaces and study some of its properties. Finally we introduce α (γ,γ')-continuous mappings and give some properties of such mappings.","grobid_abstract_attachment_id":"38348377"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"On a Class of $\\alpha_\\gamma$-Open Sets in a Topological Space","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [2367126]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":38348377,"attachmentType":"pdf"}"><img alt="First page of “On a Class of $\alpha_\gamma$-Open Sets in a Topological Space”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/38348377/mini_magick20190225-11567-1todxhc.png?1551128139" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">On a Class of $\alpha_\gamma$-Open Sets in a Topological Space</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="2367126" href="https://zakho.academia.edu/HZIbrahim"><img alt="Profile image of Hariwan Z. Ibrahim" class="ds-work-card--author-avatar" src="//a.academia-assets.com/images/s65_no_pic.png" />Hariwan Z. Ibrahim</a></div><div class="ds-work-card--detail"><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">7 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 14538401; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=14538401"; const getWorkViews = async (workId) => { const response = await fetch(worksViewsPath); if (!response.ok) { throw new Error('Failed to load work views'); } const data = await response.json(); return data.views[workId]; }; // Get the view count for the work - we send this immediately rather than waiting for // the DOM to load, so it can be available as soon as possible (but without holding up // the backend or other resource requests, because it's a bit expensive and not critical). const viewCount = await getWorkViews(workId); const updateViewCount = (viewCount) => { try { const viewCountNumber = parseInt(viewCount, 10); if (viewCountNumber === 0) { // Remove the whole views element if there are zero views. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); return; } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">In this paper, we introduce the concept of α γ -open sets as a generalization of γ-open sets in a topological space (X, τ). Using this set, we introduce α γ T 0 , α γ -T ½ , α γ T 1 , α γ T 2 , α γ D 0 , α γ D 1 and α γ D 2 spaces and study some of its properties. Finally we introduce α (γ,γ')-continuous mappings and give some properties of such mappings.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":38348377,"attachmentType":"pdf","workUrl":"https://www.academia.edu/14538401/On_a_Class_of_alpha_gamma_Open_Sets_in_a_Topological_Space"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":38348377,"attachmentType":"pdf","workUrl":"https://www.academia.edu/14538401/On_a_Class_of_alpha_gamma_Open_Sets_in_a_Topological_Space"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div><div class="ds-signup-banner-trigger-container"><div class="ds-signup-banner-trigger ds-signup-banner-trigger-control"></div></div><div class="ds-signup-banner ds-signup-banner-control"><div id="ds-signup-banner-close-button"><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--inverse"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">close</span></button></div><div class="ds-signup-banner-ctas"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><h4 class="ds2-5-heading-serif-sm">Sign up for access to the world's latest research</h4><button class="ds2-5-button ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"signup-banner"}">Sign up for free<span class="material-symbols-outlined" style="font-size: 20px" translate="no">arrow_forward</span></button></div><div class="ds-signup-banner-divider"></div><div class="ds-signup-banner-reasons"><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Get notified about relevant papers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Save papers to use in your research</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Join the discussion with peers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Track your impact</span></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. 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Further we declare the concepts of γ-α continuous mappings and γ -g α continuous mappings which coincide when the space is α-γ T-1/2. 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By using this set we introduce Pγ-T 1 2 spaces and study some of its properties. 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I El-Maghrabi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of the Egyptian Mathematical Society, 2014</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper, the notion of a-c-I-open sets in a topological space together with its corresponding interior and closure operators are introduced. Further, the concept of a-c-I-continuous functions and a-c-I-open functions are introduced and some of their basic properties are studied.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Operation approaches on α-γ-I-open sets and α-γ-I-continuous functions in topological spaces","attachmentId":92384026,"attachmentType":"pdf","work_url":"https://www.academia.edu/88406534/Operation_approaches_on_%CE%B1_%CE%B3_I_open_sets_and_%CE%B1_%CE%B3_I_continuous_functions_in_topological_spaces","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/88406534/Operation_approaches_on_%CE%B1_%CE%B3_I_open_sets_and_%CE%B1_%CE%B3_I_continuous_functions_in_topological_spaces"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="15760739" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/15760739/Bioperations_on_alpha_Open_Sets_In_Topological_Spaces">Bioperations on $\alpha$-Open Sets In Topological Spaces</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="2367126" href="https://zakho.academia.edu/HZIbrahim">Hariwan Z. Ibrahim</a></div><p class="ds-related-work--abstract ds2-5-body-sm">In this paper, we introduce the concept of $\alpha_{[\gamma, \gamma^{'}]}$-open sets in topological spaces and study some of their properties. Furthermore, we offer a new class of functions called $(\alpha_{[\gamma, \gamma^{'}]}$, $\alpha_{[\beta, \beta^{'}]})$-continuous functions and investigate their fundamental properties.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Bioperations on $\\alpha$-Open Sets In Topological Spaces","attachmentId":38788081,"attachmentType":"pdf","work_url":"https://www.academia.edu/15760739/Bioperations_on_alpha_Open_Sets_In_Topological_Spaces","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/15760739/Bioperations_on_alpha_Open_Sets_In_Topological_Spaces"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="126220223" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/126220223/On_New_Types_of_Sets_Via_%CE%B3_open_Sets_in_a_Topological_Spaces">On New Types of Sets Via γ-open Sets in (𝑎)Topological Spaces</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="327841696" href="https://independent.academia.edu/SheetalLuthra1">Sheetal Luthra</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Cubo (Temuco), 2018</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper, we introduced the notion of γ-semi-open sets and γ-P-semi-open sets in (a)topological spaces which is a set equipped with countable number of topologies. Several properties of these notions are discussed. RESUMEN En este artículo, introducimos la noción de conjuntos γ-semi-abiertos y conjuntos γ-Psemi-abiertos en espacios (a)topológicos, el cual es un conjunto dotado con una cantidad numerable de topologías. Discutimos diversas propiedades de estas nociones.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On New Types of Sets Via γ-open Sets in (𝑎)Topological Spaces","attachmentId":120130496,"attachmentType":"pdf","work_url":"https://www.academia.edu/126220223/On_New_Types_of_Sets_Via_%CE%B3_open_Sets_in_a_Topological_Spaces","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/126220223/On_New_Types_of_Sets_Via_%CE%B3_open_Sets_in_a_Topological_Spaces"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="107975355" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/107975355/MATHEMATICAL_COMMUNICATIONS_577_Math_Commun_16_2011_577_591_Operation_approach_to_%CE%B2_open_sets_and_applications">MATHEMATICAL COMMUNICATIONS 577 Math. Commun. 16(2011), 577–591. Operation approach to β-open sets and applications</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="153855937" href="https://independent.academia.edu/tahilianisanjay">sanjay tahiliani</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2008</p><p class="ds-related-work--abstract ds2-5-body-sm">Abstract. In this paper, we introduce the concept of an operation γ on a family of β-open sets denoted by βO(X) in a topological space (X, τ). Using the operation γ on βO(X), we introduce the concept of β-γ-open sets, and investigate the related topological properties. We also introduce the notion of β-γ-Ti spaces (i = 0, 1/2, 1, 2) and study some topological properties on them. Further, we introduce β-(γ, b)-continuous maps and investigate basic properties. Finally, we investigate a general operation approach to β-closed graphs of mappings.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"MATHEMATICAL COMMUNICATIONS 577 Math. Commun. 16(2011), 577–591. Operation approach to β-open sets and applications","attachmentId":106486140,"attachmentType":"pdf","work_url":"https://www.academia.edu/107975355/MATHEMATICAL_COMMUNICATIONS_577_Math_Commun_16_2011_577_591_Operation_approach_to_%CE%B2_open_sets_and_applications","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/107975355/MATHEMATICAL_COMMUNICATIONS_577_Math_Commun_16_2011_577_591_Operation_approach_to_%CE%B2_open_sets_and_applications"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="24568721" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/24568721/ON_An_Infra_%CE%B1_Open_Sets">ON An Infra-α-Open Sets</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="37717405" href="https://scribd.academia.edu/HakeemOthman">Hakeem Othman</a></div><p class="ds-related-work--abstract ds2-5-body-sm">In this paper, we define a new class of set in general topology called an infra-α-open set and we investigate fundamental properties by using this new class. The relation between infra-α-open set and other topological sets are studied. Moreover, In the light of this new definition, we also define some generalization of continuous mappings and discuss the relations between these new classes of mappings and other continuous mappings. Basic properties of these new mappings are studied and we apply these new classes to give characterization of connected space.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"ON An Infra-α-Open Sets","attachmentId":44897436,"attachmentType":"pdf","work_url":"https://www.academia.edu/24568721/ON_An_Infra_%CE%B1_Open_Sets","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/24568721/ON_An_Infra_%CE%B1_Open_Sets"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="23242910" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/23242910/On_some_maps_concerning_g_and_946_and_952_open_sets">On some maps concerning g&#946;&#952;-open sets</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="45104431" href="https://independent.academia.edu/SaeidJafari5">Saeid Jafari</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Proyecciones (Antofagasta), 2015</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper, we consider a new generalization of α-open maps via the concept of gα-closed sets which we call approximately α-open maps. We study some of its fundamental properties. It turns out that we can use this notion to obtain a new characterization of α-T 1/2 spaces.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On some maps concerning g\u0026#946;\u0026#952;-open sets","attachmentId":43725996,"attachmentType":"pdf","work_url":"https://www.academia.edu/23242910/On_some_maps_concerning_g_and_946_and_952_open_sets","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/23242910/On_some_maps_concerning_g_and_946_and_952_open_sets"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":38348377,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":38348377,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_38348377" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. 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(PDF) On a Class of $\​alpha_\​gamma$-Open Sets in a Topological Space

(PDF) On a Class of $\alpha_\gamma$-Open Sets in a Topological Space