Refining and extending the theoretical foundations of r-near topology

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, 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.