{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T02:08:06Z","timestamp":1740103686148,"version":"3.37.3"},"reference-count":30,"publisher":"Wiley","license":[{"start":{"date-parts":[[2020,5,28]],"date-time":"2020-05-28T00:00:00Z","timestamp":1590624000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100009392","name":"Prince Sattam bin Abdulaziz University","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100009392","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2020,5,28]]},"abstract":"<jats:p>Our purpose is to present the notions of a <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mi>\u03b2<\/mml:mi><\/mml:math>-<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mi>\u039b<\/mml:mi><\/mml:math>-set and a <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mi>\u03b2<\/mml:mi><\/mml:math>-<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mi>V<\/mml:mi><\/mml:math>-sets in topological space. We discuss the basic properties of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mi>\u03b2<\/mml:mi><\/mml:math>-<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mi>\u039b<\/mml:mi><\/mml:math>-sets and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mi>\u03b2<\/mml:mi><\/mml:math>-<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mi>V<\/mml:mi><\/mml:math>-sets. Also, the achievement of the topology defined by these families of sets is obtained. Finally, these results are applied to the case of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:mi>X<\/mml:mi><mml:mo>,<\/mml:mo><mml:mi>\u03c4<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math> which is the digital <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\"><mml:mi>n<\/mml:mi><\/mml:math>-space <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\"><mml:mfenced open=\"(\" close=\")\" separators=\"|\"><mml:mrow><mml:msup><mml:mrow><mml:mi>Z<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msup><mml:mo>,<\/mml:mo><mml:msup><mml:mrow><mml:mi>K<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>n<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:mfenced><\/mml:math> (cf. Section 4).<\/jats:p>","DOI":"10.1155\/2020\/4237462","type":"journal-article","created":{"date-parts":[[2020,5,28]],"date-time":"2020-05-28T23:33:24Z","timestamp":1590708804000},"page":"1-8","source":"Crossref","is-referenced-by-count":3,"title":["Some Topological Notations via Maki\u2019s \u039b-Sets"],"prefix":"10.1155","volume":"2020","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1206-9326","authenticated-orcid":true,"given":"A. A.","family":"Azzam","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Humanities, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, New Valley University, Elkharga 72511, Egypt"}]},{"given":"A. A.","family":"Nasef","sequence":"additional","affiliation":[{"name":"Department of Physics and Engineering Mathematics, Faculty of Engineering, Kafrelsheikh University, Kafrelsheikh, Egypt"}]}],"member":"311","reference":[{"key":"1","first-page":"77","volume":"12","year":"1983","journal-title":"Bulletin of the Faculty of Science. Assiut University"},{"year":"1986","key":"2","first-page":"139"},{"issue":"1070","key":"3","first-page":"89","volume":"19","year":"1970","journal-title":"Rendiconti del Circolo Matematico di Palermo"},{"key":"4","first-page":"55","volume":"22","year":"1982","journal-title":"Kyungpook Mathematical Journal"},{"key":"5","first-page":"21","volume":"21","year":"2000","journal-title":"Mem. Fac. Sci. Kochi Univ. Math."},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1023\/a:1015605426358"},{"key":"7","first-page":"1","volume":"25","year":"2003","journal-title":"Mem. Fac. Sci. Kochi Univ. 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