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AIMS Mathematics

2025, Volume 10, Issue 12: 30429-30459. doi: 10.3934/math.20251335
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Refining and extending the theoretical foundations of $ r $-near topology

  • Department of Mathematics, Faculty of Science, Çanakkale Onsekiz Mart University, Çanakkale, Türkiye
  • Published: 25 December 2025

  • MSC : 03E20, 54A05, 54D99

  • The present paper aims to refine and extend the theoretical foundations of $ r $-near topology. For this reason, it first redefines the concept of $ r $-near neighborhoods to address inconsistencies in previous studies and clarifies the relationship between $ r $-near open neighborhoods and $ r $-near closure. This study then elaborates on the fundamental properties of $ r $-near closed sets, $ r $-near interior, $ r $-near closure, and $ r $-near neighborhoods. Subsequently, it introduces four novel concepts within $ r $-near topology: $ r $-near accumulation points, $ r $-near isolated points, $ r $-near exterior points, and $ r $-near boundary points. Furthermore, this study explores some of their basic properties and provides illustrative examples regarding the aforesaid concepts. Additionally, it researches the relationships between $ r $-near interior, $ r $-near closure, and $ r $-near exterior in the $ r $-near topological spaces and their classical topological counterparts. Lastly, the study highlights the theoretical significance of $ r $-near topology and suggests potential directions for further research.

    Citation: Tuğçe Aydın. Refining and extending the theoretical foundations of $ r $-near topology[J]. AIMS Mathematics, 2025, 10(12): 30429-30459. doi: 10.3934/math.20251335

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  • The present paper aims to refine and extend the theoretical foundations of $ r $-near topology. For this reason, it first redefines the concept of $ r $-near neighborhoods to address inconsistencies in previous studies and clarifies the relationship between $ r $-near open neighborhoods and $ r $-near closure. This study then elaborates on the fundamental properties of $ r $-near closed sets, $ r $-near interior, $ r $-near closure, and $ r $-near neighborhoods. Subsequently, it introduces four novel concepts within $ r $-near topology: $ r $-near accumulation points, $ r $-near isolated points, $ r $-near exterior points, and $ r $-near boundary points. Furthermore, this study explores some of their basic properties and provides illustrative examples regarding the aforesaid concepts. Additionally, it researches the relationships between $ r $-near interior, $ r $-near closure, and $ r $-near exterior in the $ r $-near topological spaces and their classical topological counterparts. Lastly, the study highlights the theoretical significance of $ r $-near topology and suggests potential directions for further research.



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