Aydin, Tugce2026-02-032026-02-0320252473-6988https://doi.org/10.3934/math.20251335https://hdl.handle.net/20.500.12428/34412The 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, rnear closure, and r-near neighborhoods. Subsequently, it introduces four novel concepts within rnear topology: r-near accumulation points, r-near isolated points, r-near exterior points, and rnear 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.eninfo:eu-repo/semantics/openAccessnear setsr-near topologyr-near open setsr-near accumulation pointsr-near boundary pointsArticle1012304293045910.3934/math.20251335Q1WOS:0016515571000012-s2.0-105026261723Q1