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The definitions provided by the authors of the current study are offered together with a discussion of the recent advances that they have contributed. We begin with an introduction to fn-Fr#bϱN, which includes the concepts of closed and open sets. We explore characteristics in fn-βd#bϱN and fn-℮bϱN(ϱN), and provide an idea of obtained results by adding the notion of FNbϱNOS and analyzing a few of their properties in fnts. We've researched the contrasts between the derived, exterior, and frontier notions that are provided. We also looked at the ideas of 〈TbϱN〉 C-functions and Γs-segregated functions and examined and determined the traits. [ABSTRACT FROM AUTHOR]