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, $r$-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.